{"id":"n-c6e3efe87a93824d","revision":0,"first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","name":"derivative of derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-3,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":true,"rows":[[],[],["2"],["0","6"],["0","0","12"],["0","0","0","20"],["0","0","0","0","30"],["0","0","0","0","0","42"],["0","0","0","0","0","0","56"],["0","0","0","0","0","0","0","72"],["0","0","0","0","0","0","0","0","90"],["0","0","0","0","0","0","0","0","0","110"],["0","0","0","0","0","0","0","0","0","0","132"],["0","0","0","0","0","0","0","0","0","0","0","156"],["0","0","0","0","0","0","0","0","0","0","0","0","182"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","2","6","12","20","30","42","56","72","90","110","132","156","182"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171062,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-61cd49713ca3055b"]},{"id":171063,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-61cd49713ca3055b"]},{"id":171064,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-61cd49713ca3055b"]}],"region_exact":{},"incoming_edges":[{"id":"e-61cd49713ca3055b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-c6e3efe87a93824d","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"a0071d6fee302b4328d17229f234e85cb706d7c45306a7333aaaf0ac997048ab","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e9cf5b3881297005","role":"sequence","name":"derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0124e39a9d436fca","kind":"fingerprint_related","node_a_id":"n-c234358d95fc28fc","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-c234358d95fc28fc","other":{"id":"n-c234358d95fc28fc","name":"theta_plus_one of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-2"],["0","0","3"]],"card_rows":[["1"],["0","-2"],["0","0","3"],["0","0","0","-4"]],"row_sums_preview":["1","-2","3","-4","5","-6","7","-8"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["\u03b8+1"],"edges":["e-5a10774a37bfecce"],"text":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1","chain":null}},{"id":"l-023d5f811b17a4ec","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-c2ae434b523eb55c","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-c2ae434b523eb55c","other":{"id":"n-c2ae434b523eb55c","name":"theta_plus_one_squared of theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","16"],["0","0","162"]],"card_rows":[["1"],["0","16"],["0","0","162"],["0","0","0","1536"]],"row_sums_preview":["1","16","162","1536","15000","155520","1728720","20643840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-76872140ee232c50","e-a0adc9ac4840aa41"],"text":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-07cfaf3bb845fe4c","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-8db75f631150c8cb","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-8db75f631150c8cb","other":{"id":"n-8db75f631150c8cb","name":"derivative of veronese_section_2_1 of A natural number operator.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],[]],"card_rows":[["7"],["8"],["0","22"],["0","24"]],"row_sums_preview":["0","0","0","7","8","22","24","45"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f4c1dee76f5a68b2","steps":["Veronese 2 (j=1)","D"],"edges":["e-611716aca3d048f2","e-c21c68ea2ad83f3f"],"text":"A132774 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A132774 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}},{"id":"l-0b2e862028a4cdc3","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-948d28ca3d2a34ad","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-948d28ca3d2a34ad","other":{"id":"n-948d28ca3d2a34ad","name":"derivative of theta_plus_one_squared of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-4"],["0","18"]],"card_rows":[["-4"],["0","18"],["0","0","-48"],["0","0","0","100"]],"row_sums_preview":["0","-4","18","-48","100","-180","294","-448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-d58e809d3242b4fa","e-5b6da8ebe7a9a94f"],"text":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-0c09874272c51c24","kind":"fingerprint_related","node_a_id":"n-0fadb6a502ab98be","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0fadb6a502ab98be","other":{"id":"n-0fadb6a502ab98be","name":"theta_plus_one of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-2"],["0","0","6"]],"card_rows":[["1"],["0","-2"],["0","0","6"],["0","0","0","-24"]],"row_sums_preview":["1","-2","6","-24","120","-720","5040","-40320"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["\u03b8+1"],"edges":["e-19f259e2ece869e6"],"text":"basis-map coefficients of A021012 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A021012 \u2192 \u03b8+1","chain":null}},{"id":"l-0cb6e18dedaf19fc","kind":"fingerprint_related","node_a_id":"n-5118548b67eae4b6","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-5118548b67eae4b6","other":{"id":"n-5118548b67eae4b6","name":"theta_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","8"],["0","0","21"]],"card_rows":[["1"],["0","8"],["0","0","21"],["0","0","0","40"]],"row_sums_preview":["1","8","21","40","65","96","133","176"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["\u03b8+1"],"edges":["e-5734274d793ba02f"],"text":"A131431 \u2192 \u03b8+1"},"headline":"A131431 \u2192 \u03b8+1","chain":null}},{"id":"l-0cfb8dc3715f31af","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-94851bacfe7c4fa6","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-94851bacfe7c4fa6","other":{"id":"n-94851bacfe7c4fa6","name":"derivative of derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["18"]],"card_rows":[["18"],["0","96"],["0","0","300"],["0","0","0","720"]],"row_sums_preview":["0","0","18","96","300","720","1470","2688"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D","D"],"edges":["e-06738ea5bb413735","e-864c354a3903aafb","e-6805e0422615357f"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D \u2192 D"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D \u2192 D","chain":null}},{"id":"l-0d6646d472ef4afc","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-b356dda7d5147193","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b356dda7d5147193","other":{"id":"n-b356dda7d5147193","name":"reversal of theta_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8"],["21"]],"card_rows":[["1"],["8"],["21"],["40"]],"row_sums_preview":["1","8","21","40","65","96","133","176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["\u03b8+1","reverse"],"edges":["e-5734274d793ba02f","e-95a8d3a4c22397df"],"text":"A131431 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A131431 \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-0e50b5a9758cf1e2","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-f57c653f691514a0","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-f57c653f691514a0","other":{"id":"n-f57c653f691514a0","name":"theta_plus_one of derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","12"],["0","0","36"],["0","0","0","80"]],"row_sums_preview":["0","0","2","12","36","80","150","252"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["D","\u03b8+1"],"edges":["e-1aec373ef809ab0b","e-c45d54ebf132bae2"],"text":"A127648 \u2192 D \u2192 \u03b8+1"},"headline":"A127648 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-1151cf61da6e5697","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-5963d86662c046c4","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-5963d86662c046c4","other":{"id":"n-5963d86662c046c4","name":"theta_plus_one_squared of derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","8"]],"card_rows":[["1"],["0","8"],["0","0","27"],["0","0","0","64"]],"row_sums_preview":["0","1","8","27","64","125","216","343"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-198d139196a8bb12","e-0c371c2a79ae53e1"],"text":"basis-map coefficients of A026729 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A026729 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-12f7f71465ed8f5b","kind":"fingerprint_related","node_a_id":"n-665a2c7873826b72","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-665a2c7873826b72","other":{"id":"n-665a2c7873826b72","name":"gamma_polynomial of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["2"]],"card_rows":[["1"],["1"],["2"],["6"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["\u03b3"],"edges":["e-d8c50adb346fbffe"],"text":"A196347 \u2192 \u03b3"},"headline":"A196347 \u2192 \u03b3","chain":null}},{"id":"l-1327c50b1347c182","kind":"fingerprint_related","node_a_id":"n-476b54f99a53261d","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-476b54f99a53261d","other":{"id":"n-476b54f99a53261d","name":"gamma_polynomial of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["6"]],"card_rows":[["1"],["2"],["6"],["24"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["\u03b3"],"edges":["e-db870fb061b168b8"],"text":"A156992 \u2192 \u03b3"},"headline":"A156992 \u2192 \u03b3","chain":null}},{"id":"l-159b45bffb696da8","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-19c0690cfb4c0a03","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-19c0690cfb4c0a03","other":{"id":"n-19c0690cfb4c0a03","name":"theta_plus_one of derivative of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["0","24"]],"card_rows":[["3"],["0","24"],["0","0","90"],["0","0","0","240"]],"row_sums_preview":["0","3","24","90","240","525","1008","1764"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["D","\u03b8+1"],"edges":["e-353b5c171e173f11","e-e1368666b89da845"],"text":"A127773 \u2192 D \u2192 \u03b8+1"},"headline":"A127773 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-1731d9d99870ba22","kind":"fingerprint_related","node_a_id":"n-6fea5aff4f44fa54","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-6fea5aff4f44fa54","other":{"id":"n-6fea5aff4f44fa54","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","0","1"]],"card_rows":[["1"],["0","1"],["0","0","1"],["0","0","0","1"]],"row_sums_preview":["1","1","1","1","1","1","1","1"],"regions":["(-inf, 0]","[0, +inf)","[-1, 0]"],"region_exact":{},"oeis_ids":["A023531"],"derivation":{"root":"n-6fea5aff4f44fa54","steps":[],"edges":[],"text":"A023531"},"headline":"A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","chain":null}},{"id":"l-1821367db9466dec","kind":"fingerprint_related","node_a_id":"n-0efe468f23ab158d","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0efe468f23ab158d","other":{"id":"n-0efe468f23ab158d","name":"theta_plus_one of derivative of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["0","4"]],"card_rows":[["-1"],["0","4"],["0","0","-9"],["0","0","0","16"]],"row_sums_preview":["0","-1","4","-9","16","-25","36","-49"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["D","\u03b8+1"],"edges":["e-7b1079618314be79","e-c40fa60a569f593b"],"text":"basis-map coefficients of A030528, shifted \u2192 D \u2192 \u03b8+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-187a2cfcf6f6cebe","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-d4acb1e46d2a5da9","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-d4acb1e46d2a5da9","other":{"id":"n-d4acb1e46d2a5da9","name":"derivative of theta_plus_one of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["0","36"]],"card_rows":[["6"],["0","36"],["0","0","120"],["0","0","0","300"]],"row_sums_preview":["0","6","36","120","300","630","1176","2016"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["\u03b8+1","D"],"edges":["e-9b1f1f612dc2ed52","e-b0500811913c926d"],"text":"A127773 \u2192 \u03b8+1 \u2192 D"},"headline":"A127773 \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-19562e8c60bf9139","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-beba9729523d3265","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-beba9729523d3265","other":{"id":"n-beba9729523d3265","name":"theta_plus_one_squared of theta_plus_one_squared of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","81"],["0","0","0","256"],["0","0","0","0","625"]],"card_rows":[["0","0","81"],["0","0","0","256"],["0","0","0","0","625"],["0","0","0","0","0","1296"]],"row_sums_preview":["81","256","625","1296","2401","4096","6561","10000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-c527a11d264443d6","e-e41b3479001b9202"],"text":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-19900821fce68eef","kind":"fingerprint_related","node_a_id":"n-e75bc09e82ca4254","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e75bc09e82ca4254","other":{"id":"n-e75bc09e82ca4254","name":"derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["0","2"],["0","0","0","4"]],"card_rows":[["0","2"],["0","0","0","4"],["0","0","0","0","0","6"],["0","0","0","0","0","0","0","8"]],"row_sums_preview":["0","2","4","6","8","10","12","14"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["D"],"edges":["e-521a2a476289c028"],"text":"basis-map coefficients of A297191, shifted \u2192 D"},"headline":"basis-map coefficients of A297191, shifted \u2192 D","chain":null}},{"id":"l-1b6ffcee2d4674fb","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-b729867594f9a842","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b729867594f9a842","other":{"id":"n-b729867594f9a842","name":"theta_plus_one_squared of derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","24"],["0","0","108"],["0","0","0","320"]],"row_sums_preview":["0","0","2","24","108","320","750","1512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-1aec373ef809ab0b","e-9aae1aa5e2e67ab1"],"text":"A127648 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A127648 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-1bef920a96302487","kind":"fingerprint_related","node_a_id":"n-5248ff3010d9702d","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-5248ff3010d9702d","other":{"id":"n-5248ff3010d9702d","name":"reversal of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["6"]],"card_rows":[["1"],["3"],["6"],["10"]],"row_sums_preview":["1","3","6","10","15","21","28","36"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["reverse"],"edges":["e-626d66bf93a4e3ed"],"text":"A127773 \u2192 reverse"},"headline":"A127773 \u2192 reverse","chain":null}},{"id":"l-1f02ad583bc4afc7","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-0925e236d9af8f61","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0925e236d9af8f61","other":{"id":"n-0925e236d9af8f61","name":"reversal of theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2"],["3"]],"card_rows":[["1"],["-2"],["3"],["-4"]],"row_sums_preview":["1","-2","3","-4","5","-6","7","-8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["\u03b8+1","reverse"],"edges":["e-1d47beb63d4711b8","e-95cf5fe7ee4727f0"],"text":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 reverse"},"headline":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-201a654f92b926b3","kind":"fingerprint_related","node_a_id":"n-b9533c4d23e51af7","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b9533c4d23e51af7","other":{"id":"n-b9533c4d23e51af7","name":"theta_plus_one_squared of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","8"]],"card_rows":[["1"],["0","8"],["0","0","27"],["0","0","0","64"]],"row_sums_preview":["0","1","8","27","64","125","216","343"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["(\u03b8+1)\u00b2"],"edges":["e-10cf23660c74d6bc"],"text":"A127648 \u2192 (\u03b8+1)\u00b2"},"headline":"A127648 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-20c5402200ba94b9","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-59838c4700a38774","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-59838c4700a38774","other":{"id":"n-59838c4700a38774","name":"theta_plus_one of theta_plus_one_squared of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","16"]],"card_rows":[["1"],["0","16"],["0","0","81"],["0","0","0","256"]],"row_sums_preview":["0","1","16","81","256","625","1296","2401"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-10cf23660c74d6bc","e-5d968abd58622293"],"text":"A127648 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"A127648 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-27648f7227822f76","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-17978f7c6efc53df","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-17978f7c6efc53df","other":{"id":"n-17978f7c6efc53df","name":"theta_plus_one_squared of theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","0","81"],["0","0","0","0","625"]],"card_rows":[["1"],["0","0","81"],["0","0","0","0","625"],["0","0","0","0","0","0","2401"]],"row_sums_preview":["1","81","625","2401","6561","14641","28561","50625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-5ac7ff393747d6fc","e-26acb929b519972d"],"text":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-2772bb59fea09448","kind":"fingerprint_related","node_a_id":"n-f71a45e9f51781db","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-f71a45e9f51781db","other":{"id":"n-f71a45e9f51781db","name":"gamma_polynomial of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4"],["10"]],"card_rows":[["1"],["4"],["10"],["20"]],"row_sums_preview":["1","4","10","20","35","56","84","120"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["\u03b3"],"edges":["e-cf285083c5135dab"],"text":"A178820 \u2192 \u03b3"},"headline":"A178820 \u2192 \u03b3","chain":null}},{"id":"l-2948f2d3164d0d78","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-34d912f2cc4b559b","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-34d912f2cc4b559b","other":{"id":"n-34d912f2cc4b559b","name":"derivative of theta_plus_one_squared of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["8"]],"card_rows":[["8"],["0","54"],["0","0","192"],["0","0","0","500"]],"row_sums_preview":["0","0","8","54","192","500","1080","2058"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-10cf23660c74d6bc","e-5e35cf260bbf93fd"],"text":"A127648 \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"A127648 \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-2998b5f6ccf4cfa7","kind":"fingerprint_related","node_a_id":"n-690bbd876204988f","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-690bbd876204988f","other":{"id":"n-690bbd876204988f","name":"derivative of theta_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","6"],["0","0","12"],["0","0","0","20"]],"card_rows":[["0","6"],["0","0","12"],["0","0","0","20"],["0","0","0","0","30"]],"row_sums_preview":["6","12","20","30","42","56","72","90"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["\u03b8+1","D"],"edges":["e-972f6eabe8338070","e-5c37f65ab508c84f"],"text":"basis-map coefficients of A112227 \u2192 \u03b8+1 \u2192 D"},"headline":"basis-map coefficients of A112227 \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-2c717b7dbf410fd1","kind":"fingerprint_related","node_a_id":"n-dabe0dfba915bf52","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-dabe0dfba915bf52","other":{"id":"n-dabe0dfba915bf52","name":"reversal of Triangle read by rows: row n consists of n zeros followed by n+1.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["3"]],"card_rows":[["1"],["2"],["3"],["4"]],"row_sums_preview":["1","2","3","4","5","6","7","8"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["\u03b3"],"edges":["e-6239f8c0a2c99666"],"text":"A003506 \u2192 \u03b3"},"headline":"A003506 \u2192 \u03b3","chain":null}},{"id":"l-2d78be3150df1ee4","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-11817a9d1f935606","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-11817a9d1f935606","other":{"id":"n-11817a9d1f935606","name":"reversal of hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["12"]],"card_rows":[["1"],["3"],["12"],["60"]],"row_sums_preview":["1","3","12","60","360","2520","20160","181440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f7eaad3557182df","steps":["reverse"],"edges":["e-fbcfcbb1806b4823"],"text":"Hadamard-factor parent of A127773 \u2192 reverse"},"headline":"Hadamard-factor parent of A127773 \u2192 reverse","chain":null}},{"id":"l-305cb79cd0b974ee","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-128b2bff3dcd450b","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-128b2bff3dcd450b","other":{"id":"n-128b2bff3dcd450b","name":"theta_plus_one_squared of theta_plus_one_squared of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","64"],["0","0","567"]],"card_rows":[["1"],["0","64"],["0","0","567"],["0","0","0","2560"]],"row_sums_preview":["1","64","567","2560","8125","20736","45619","90112"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-792e4f12e133c3d5","e-29b3ac1efd0dc2a4"],"text":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-3326bb8dab16d334","kind":"fingerprint_related","node_a_id":"n-511cbcf068838141","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-511cbcf068838141","other":{"id":"n-511cbcf068838141","name":"theta_plus_one_squared of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","16"],["0","0","63"]],"card_rows":[["1"],["0","16"],["0","0","63"],["0","0","0","160"]],"row_sums_preview":["1","16","63","160","325","576","931","1408"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["(\u03b8+1)\u00b2"],"edges":["e-792e4f12e133c3d5"],"text":"A131431 \u2192 (\u03b8+1)\u00b2"},"headline":"A131431 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-336f65a48a59608e","kind":"fingerprint_related","node_a_id":"n-7c6e1d5e4a537ecf","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7c6e1d5e4a537ecf","other":{"id":"n-7c6e1d5e4a537ecf","name":"gamma_polynomial of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["2"],["2"]],"card_rows":[["2"],["2"],["2"],["2"]],"row_sums_preview":["2","2","2","2","2","2","2","2"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["\u03b3"],"edges":["e-9171c77362f53bc9"],"text":"A028326 \u2192 \u03b3"},"headline":"A028326 \u2192 \u03b3","chain":null}},{"id":"l-3509d4e541390073","kind":"fingerprint_related","node_a_id":"n-67e8e5fe38a23ea9","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-67e8e5fe38a23ea9","other":{"id":"n-67e8e5fe38a23ea9","name":"reversal of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1"],["1"]],"card_rows":[["1"],["-1"],["1"],["-1"]],"row_sums_preview":["1","-1","1","-1","1","-1","1","-1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["reverse"],"edges":["e-0bd5b74a5428669b"],"text":"basis-map coefficients of A030528, shifted \u2192 reverse"},"headline":"basis-map coefficients of A030528, shifted \u2192 reverse","chain":null}},{"id":"l-35e754b1308c0672","kind":"fingerprint_related","node_a_id":"n-47e7df75357651f4","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-47e7df75357651f4","other":{"id":"n-47e7df75357651f4","name":"reversal of theta_plus_one of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2"],["3"]],"card_rows":[["1"],["-2"],["3"],["-4"]],"row_sums_preview":["1","-2","3","-4","5","-6","7","-8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["\u03b8+1","reverse"],"edges":["e-5a10774a37bfecce","e-5314b6b45ce63068"],"text":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 reverse"},"headline":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-4097abfc31638f00","kind":"fingerprint_related","node_a_id":"n-169539049113178e","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-169539049113178e","other":{"id":"n-169539049113178e","name":"derivative of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["0","12"]],"card_rows":[["3"],["0","12"],["0","0","30"],["0","0","0","60"]],"row_sums_preview":["0","3","12","30","60","105","168","252"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["D"],"edges":["e-353b5c171e173f11"],"text":"A127773 \u2192 D"},"headline":"A127773 \u2192 D","chain":null}},{"id":"l-41464ab233e1b2d0","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-9d32d60c58bed829","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-9d32d60c58bed829","other":{"id":"n-9d32d60c58bed829","name":"derivative of theta_plus_one of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-2"],["0","12"]],"card_rows":[["-2"],["0","12"],["0","0","-72"],["0","0","0","480"]],"row_sums_preview":["0","-2","12","-72","480","-3600","30240","-282240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["\u03b8+1","D"],"edges":["e-19f259e2ece869e6","e-002561c69192f35f"],"text":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 D"},"headline":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-43989a605ac5e850","kind":"fingerprint_related","node_a_id":"n-356713338f14325b","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-356713338f14325b","other":{"id":"n-356713338f14325b","name":"Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","2"],["0","0","2"]],"card_rows":[["1"],["0","2"],["0","0","2"],["0","0","0","2"]],"row_sums_preview":["1","2","2","2","2","2","2","2"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A136521"],"derivation":{"root":"n-356713338f14325b","steps":[],"edges":[],"text":"A136521"},"headline":"A136521: Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.","chain":null}},{"id":"l-43aafc77e5e2dad7","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-68128ecab2720b7c","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-68128ecab2720b7c","other":{"id":"n-68128ecab2720b7c","name":"reversal of theta_plus_one_squared of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8"],["36"]],"card_rows":[["1"],["8"],["36"],["128"]],"row_sums_preview":["1","8","36","128","400","1152","3136","8192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-c89dffe6ace51b80","e-580fa37ebb3c55ee"],"text":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-44aff7780e0a8fdb","kind":"fingerprint_related","node_a_id":"n-b961cc69f2b3d539","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b961cc69f2b3d539","other":{"id":"n-b961cc69f2b3d539","name":"veronese_section_2_0 of A natural number operator.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["0","5"]],"card_rows":[["1"],["2"],["0","5"],["0","6"]],"row_sums_preview":["1","2","5","6","9","10","13","14"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f4c1dee76f5a68b2","steps":["Veronese 2"],"edges":["e-bf26e3fe4e7c28ab"],"text":"A132774 \u2192 Veronese 2"},"headline":"A132774 \u2192 Veronese 2","chain":null}},{"id":"l-44dffa6d30ca5db0","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-ca232f62495e320c","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ca232f62495e320c","other":{"id":"n-ca232f62495e320c","name":"reversal of theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4"],["18"]],"card_rows":[["1"],["4"],["18"],["96"]],"row_sums_preview":["1","4","18","96","600","4320","35280","322560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-76872140ee232c50","e-858b3d40e720983c"],"text":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-470b1fb4a57c642c","kind":"fingerprint_related","node_a_id":"n-ee4b2d952a0ea7a9","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ee4b2d952a0ea7a9","other":{"id":"n-ee4b2d952a0ea7a9","name":"gamma_polynomial of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6"],["21"]],"card_rows":[["1"],["6"],["21"],["56"]],"row_sums_preview":["1","6","21","56","126","252","462","792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["\u03b3"],"edges":["e-1b976fb6ae15a002"],"text":"A178822 \u2192 \u03b3"},"headline":"A178822 \u2192 \u03b3","chain":null}},{"id":"l-4767235f37444c50","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-c35b30c6d58f99b4","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-c35b30c6d58f99b4","other":{"id":"n-c35b30c6d58f99b4","name":"theta_plus_one_squared of derivative of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","16"]],"card_rows":[["1"],["0","16"],["0","0","162"],["0","0","0","1536"]],"row_sums_preview":["0","1","16","162","1536","15000","155520","1728720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-789a32df93c7cdb5","e-1c6e0215988361a6"],"text":"A136572 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A136572 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-4a621be6d8a5aa0a","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-991280066819b911","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-991280066819b911","other":{"id":"n-991280066819b911","name":"theta_plus_one of veronese_section_2_0 of A natural number operator.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["0","10"]],"card_rows":[["1"],["2"],["0","10"],["0","12"]],"row_sums_preview":["1","2","10","12","27","30","52","56"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f4c1dee76f5a68b2","steps":["Veronese 2","\u03b8+1"],"edges":["e-bf26e3fe4e7c28ab","e-1b6a8dd192228785"],"text":"A132774 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A132774 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}},{"id":"l-4b01bcbadf92d7f3","kind":"fingerprint_related","node_a_id":"n-6b40f6aca669811e","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-6b40f6aca669811e","other":{"id":"n-6b40f6aca669811e","name":"theta_plus_one of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","4"]],"card_rows":[["1"],["0","4"],["0","0","9"],["0","0","0","16"]],"row_sums_preview":["0","1","4","9","16","25","36","49"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["\u03b8+1"],"edges":["e-0695c41826b28137"],"text":"A132440 \u2192 \u03b8+1"},"headline":"A132440 \u2192 \u03b8+1","chain":null}},{"id":"l-4b5f7d5b4b1b77c1","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-3f0002a715c9c816","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-3f0002a715c9c816","other":{"id":"n-3f0002a715c9c816","name":"theta_plus_one_squared of veronese_section_2_1 of A natural number operator.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["4"]],"card_rows":[["3"],["4"],["0","28"],["0","32"]],"row_sums_preview":["0","3","4","28","32","99","108","240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f4c1dee76f5a68b2","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-611716aca3d048f2","e-b292873dceb69363"],"text":"A132774 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A132774 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-4d2dc15ca548b216","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-e8ecdd2ad0c784ae","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e8ecdd2ad0c784ae","other":{"id":"n-e8ecdd2ad0c784ae","name":"theta_plus_one of derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["0","36"]],"card_rows":[["4"],["0","36"],["0","0","144"],["0","0","0","400"]],"row_sums_preview":["0","4","36","144","400","900","1764","3136"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D","\u03b8+1"],"edges":["e-06738ea5bb413735","e-864c354a3903aafb","e-8667da44eef44dd4"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D \u2192 \u03b8+1"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-4d4eaffd5fba8e46","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-22baeff6980f38d6","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-22baeff6980f38d6","other":{"id":"n-22baeff6980f38d6","name":"derivative of theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-2"],["0","6"]],"card_rows":[["-2"],["0","6"],["0","0","-12"],["0","0","0","20"]],"row_sums_preview":["0","-2","6","-12","20","-30","42","-56"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["\u03b8+1","D"],"edges":["e-1d47beb63d4711b8","e-6a526bf25224ca7a"],"text":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 D"},"headline":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-4ee20a28c8fc7d9f","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-7b27b0374667b1fe","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7b27b0374667b1fe","other":{"id":"n-7b27b0374667b1fe","name":"reversal of theta_plus_one of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2"],["6"]],"card_rows":[["1"],["-2"],["6"],["-24"]],"row_sums_preview":["1","-2","6","-24","120","-720","5040","-40320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["\u03b8+1","reverse"],"edges":["e-19f259e2ece869e6","e-1c1788adbafc4a5d"],"text":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 reverse"},"headline":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-4f13c8265918b84f","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-603e83c50178aea5","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-603e83c50178aea5","other":{"id":"n-603e83c50178aea5","name":"derivative of theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["0","12"]],"card_rows":[["2"],["0","12"],["0","0","72"],["0","0","0","480"]],"row_sums_preview":["0","2","12","72","480","3600","30240","282240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["\u03b8+1","D"],"edges":["e-4dc8adfab569fa5c","e-f7c537b129f7cd86"],"text":"A136572 \u2192 \u03b8+1 \u2192 D"},"headline":"A136572 \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-5183c904a8835cba","kind":"fingerprint_related","node_a_id":"n-eae3ff6a85d45ca1","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-eae3ff6a85d45ca1","other":{"id":"n-eae3ff6a85d45ca1","name":"theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-2"],["0","0","3"]],"card_rows":[["1"],["0","-2"],["0","0","3"],["0","0","0","-4"]],"row_sums_preview":["1","-2","3","-4","5","-6","7","-8"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["\u03b8+1"],"edges":["e-1d47beb63d4711b8"],"text":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1","chain":null}},{"id":"l-537f4343fbb02b27","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-012af5ecd2e006a0","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-012af5ecd2e006a0","other":{"id":"n-012af5ecd2e006a0","name":"theta_plus_one_squared of theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-8"],["0","0","27"]],"card_rows":[["1"],["0","-8"],["0","0","27"],["0","0","0","-64"]],"row_sums_preview":["1","-8","27","-64","125","-216","343","-512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-23b8fb76a1e0b114","e-9f243f8f425902bd"],"text":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-548dfe681637b264","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-42ea41de1e9db0d1","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-42ea41de1e9db0d1","other":{"id":"n-42ea41de1e9db0d1","name":"derivative of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["0","18"]],"card_rows":[["4"],["0","18"],["0","0","48"],["0","0","0","100"]],"row_sums_preview":["0","4","18","48","100","180","294","448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-3411eba2e020f0d3","e-39ec0f2734c8cc36"],"text":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-5494c18adee5a5d0","kind":"fingerprint_related","node_a_id":"n-85a52a53f57066b8","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-85a52a53f57066b8","other":{"id":"n-85a52a53f57066b8","name":"theta_plus_one_squared of derivative of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["0","8"]],"card_rows":[["-1"],["0","8"],["0","0","-27"],["0","0","0","64"]],"row_sums_preview":["0","-1","8","-27","64","-125","216","-343"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-7b1079618314be79","e-d8aa97deac6ba740"],"text":"basis-map coefficients of A030528, shifted \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A030528, shifted \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-574d24fd3d12ed98","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-ce19848a4c207b69","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ce19848a4c207b69","other":{"id":"n-ce19848a4c207b69","name":"reversal of theta_plus_one_squared of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9"],["16"],["25"]],"card_rows":[["9"],["16"],["25"],["36"]],"row_sums_preview":["9","16","25","36","49","64","81","100"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-c527a11d264443d6","e-d2d4e9da18dd8852"],"text":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-5847cb4b4e6aec81","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-b9d88caf79d58115","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b9d88caf79d58115","other":{"id":"n-b9d88caf79d58115","name":"derivative of veronese_section_2_0 of A natural number operator.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["5"]],"card_rows":[["5"],["6"],["0","18"],["0","20"]],"row_sums_preview":["0","0","5","6","18","20","39","42"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f4c1dee76f5a68b2","steps":["Veronese 2","D"],"edges":["e-bf26e3fe4e7c28ab","e-6d123b5b7d07cd88"],"text":"A132774 \u2192 Veronese 2 \u2192 D"},"headline":"A132774 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-59760c8411987937","kind":"fingerprint_related","node_a_id":"n-cf1765de6ac2bd4b","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-cf1765de6ac2bd4b","other":{"id":"n-cf1765de6ac2bd4b","name":"basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-1"],["0","0","2"]],"card_rows":[["1"],["0","-1"],["0","0","2"],["0","0","0","-6"]],"row_sums_preview":["1","-1","2","-6","24","-120","720","-5040"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":[],"edges":[],"text":"basis-map coefficients of A021012"},"headline":"basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","chain":null}},{"id":"l-5cf11df439da3c09","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-c12e84f401a56acc","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-c12e84f401a56acc","other":{"id":"n-c12e84f401a56acc","name":"reversal of Triangle read by rows, (n-1) zeros followed by 2^n - 1.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["7"]],"card_rows":[["1"],["3"],["7"],["15"]],"row_sums_preview":["1","3","7","15","31","63","127","255"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6394a556f7ad3cd4","steps":["reverse"],"edges":["e-418bb1e9b37f268d"],"text":"A131436 \u2192 reverse"},"headline":"A131436 \u2192 reverse","chain":null}},{"id":"l-5e267b508eec4281","kind":"fingerprint_related","node_a_id":"n-7807fcac430a0b13","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7807fcac430a0b13","other":{"id":"n-7807fcac430a0b13","name":"reversal of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1"],["2"]],"card_rows":[["1"],["-1"],["2"],["-6"]],"row_sums_preview":["1","-1","2","-6","24","-120","720","-5040"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["reverse"],"edges":["e-c98ee7dbfb9a2305"],"text":"basis-map coefficients of A021012 \u2192 reverse"},"headline":"basis-map coefficients of A021012 \u2192 reverse","chain":null}},{"id":"l-619b0ce70ce7a9f0","kind":"fingerprint_related","node_a_id":"n-2c108d11aae26139","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-2c108d11aae26139","other":{"id":"n-2c108d11aae26139","name":"veronese parent of Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["0","1"],["0","0","-1"],["0","0","0","1"]],"card_rows":[["0","1"],["0","0","-1"],["0","0","0","1"],["0","0","0","0","-1"]],"row_sums_preview":["1","-1","1","-1","1","-1","1","-1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2c108d11aae26139","steps":[],"edges":[],"text":"Veronese parent of A003982"},"headline":"veronese parent of Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...","chain":null}},{"id":"l-66d9b7c2584efb38","kind":"fingerprint_related","node_a_id":"n-d2e9afed41ff1350","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-d2e9afed41ff1350","other":{"id":"n-d2e9afed41ff1350","name":"theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","4"],["0","0","18"]],"card_rows":[["1"],["0","4"],["0","0","18"],["0","0","0","96"]],"row_sums_preview":["1","4","18","96","600","4320","35280","322560"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["(\u03b8+1)\u00b2"],"edges":["e-76872140ee232c50"],"text":"A136572 \u2192 (\u03b8+1)\u00b2"},"headline":"A136572 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-67cfeebd5b49aae5","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-ee3d16617b515703","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ee3d16617b515703","other":{"id":"n-ee3d16617b515703","name":"theta_plus_one of derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["0","28"]],"card_rows":[["4"],["0","28"],["0","0","90"],["0","0","0","208"]],"row_sums_preview":["0","4","28","90","208","400","684","1078"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D","\u03b8+1"],"edges":["e-7e9b074076a3cdbe","e-b67ac9eae4936557"],"text":"A131431 \u2192 D \u2192 \u03b8+1"},"headline":"A131431 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-685e7e4319b160d7","kind":"fingerprint_related","node_a_id":"n-4846a7e933633245","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-4846a7e933633245","other":{"id":"n-4846a7e933633245","name":"theta_plus_one of derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["0","4"]],"card_rows":[["-1"],["0","4"],["0","0","-9"],["0","0","0","16"]],"row_sums_preview":["0","-1","4","-9","16","-25","36","-49"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["D","\u03b8+1"],"edges":["e-c22d2d4fa11fd8cc","e-51c06a3356407ccc"],"text":"basis-map coefficients of A026729, shifted \u2192 D \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-69e693c4bb314e7a","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-c959fc01c927f962","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-c959fc01c927f962","other":{"id":"n-c959fc01c927f962","name":"reversal of theta_plus_one_squared of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12"],["54"]],"card_rows":[["1"],["12"],["54"],["160"]],"row_sums_preview":["1","12","54","160","375","756","1372","2304"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-e78403b3b87fde04","e-dab45cd0ef822dfb"],"text":"A127773 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"A127773 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-6a2a96d98c76e47f","kind":"fingerprint_related","node_a_id":"n-19a1b24e85aca2bf","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-19a1b24e85aca2bf","other":{"id":"n-19a1b24e85aca2bf","name":"derivative of theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["0","6"]],"card_rows":[["2"],["0","6"],["0","0","12"],["0","0","0","20"]],"row_sums_preview":["0","2","6","12","20","30","42","56"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["D"],"edges":["e-cad3bf5dcb418d6b"],"text":"image under \u03b8+1 \u2192 D"},"headline":"image under \u03b8+1 \u2192 D","chain":null}},{"id":"l-6ad5d1330345837e","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-0bb49218c41377a2","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0bb49218c41377a2","other":{"id":"n-0bb49218c41377a2","name":"derivative of derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["14"]],"card_rows":[["14"],["0","60"],["0","0","156"],["0","0","0","320"]],"row_sums_preview":["0","0","14","60","156","320","570","924"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D","D"],"edges":["e-7e9b074076a3cdbe","e-196a8874b4dc026e"],"text":"A131431 \u2192 D \u2192 D"},"headline":"A131431 \u2192 D \u2192 D","chain":null}},{"id":"l-6b2621602d48e818","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-1b9a6f7f2a208e06","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-1b9a6f7f2a208e06","other":{"id":"n-1b9a6f7f2a208e06","name":"theta_plus_one of veronese_section_2_1 of A natural number operator.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["4"]],"card_rows":[["3"],["4"],["0","14"],["0","16"]],"row_sums_preview":["0","3","4","14","16","33","36","60"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f4c1dee76f5a68b2","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-611716aca3d048f2","e-bef7d6eb81d01d29"],"text":"A132774 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A132774 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-6b704ab0c8508dde","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-9d26a1d9f476e370","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-9d26a1d9f476e370","other":{"id":"n-9d26a1d9f476e370","name":"derivative of theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["0","6"],["0","0","0","20"]],"card_rows":[["0","6"],["0","0","0","20"],["0","0","0","0","0","42"],["0","0","0","0","0","0","0","72"]],"row_sums_preview":["0","6","20","42","72","110","156","210"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["\u03b8+1","D"],"edges":["e-5ff024f1a04a7dc8","e-8dc84ba4bcb471d9"],"text":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1 \u2192 D"},"headline":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-6ca302e2114862b6","kind":"fingerprint_related","node_a_id":"n-3890202342120839","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-3890202342120839","other":{"id":"n-3890202342120839","name":"gamma_polynomial of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["3"]],"card_rows":[["1"],["1"],["3"],["15"]],"row_sums_preview":["1","1","3","15","105","945","10395","135135"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["\u03b3"],"edges":["e-a9c75dc8bc8e63f9"],"text":"A085881 \u2192 \u03b3"},"headline":"A085881 \u2192 \u03b3","chain":null}},{"id":"l-6ea84af34664b683","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-431b252df4381a25","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-431b252df4381a25","other":{"id":"n-431b252df4381a25","name":"derivative of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-4"],["0","18"]],"card_rows":[["-4"],["0","18"],["0","0","-48"],["0","0","0","100"]],"row_sums_preview":["0","-4","18","-48","100","-180","294","-448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-23b8fb76a1e0b114","e-3adf85f268bfbc93"],"text":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-72a2234d9ff5f7b2","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-2001b14faee9cb34","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-2001b14faee9cb34","other":{"id":"n-2001b14faee9cb34","name":"derivative of derivative of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["12"]],"card_rows":[["12"],["0","60"],["0","0","180"],["0","0","0","420"]],"row_sums_preview":["0","0","12","60","180","420","840","1512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["D","D"],"edges":["e-353b5c171e173f11","e-8a2066c53b36e4cd"],"text":"A127773 \u2192 D \u2192 D"},"headline":"A127773 \u2192 D \u2192 D","chain":null}},{"id":"l-7391cb93fbbcf51d","kind":"fingerprint_related","node_a_id":"n-c6bc14b4a73b28c8","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-c6bc14b4a73b28c8","other":{"id":"n-c6bc14b4a73b28c8","name":"reversal of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1"],["1"]],"card_rows":[["1"],["-1"],["1"],["-1"]],"row_sums_preview":["1","-1","1","-1","1","-1","1","-1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["reverse"],"edges":["e-3e96bc7f546c1ca7"],"text":"basis-map coefficients of A026729, shifted \u2192 reverse"},"headline":"basis-map coefficients of A026729, shifted \u2192 reverse","chain":null}},{"id":"l-742c538eab277b09","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-7cf6484b1b8b499b","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7cf6484b1b8b499b","other":{"id":"n-7cf6484b1b8b499b","name":"theta_plus_one of derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","4"]],"card_rows":[["1"],["0","4"],["0","0","9"],["0","0","0","16"]],"row_sums_preview":["0","1","4","9","16","25","36","49"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["D","\u03b8+1"],"edges":["e-198d139196a8bb12","e-b2bf6fb3a5101c51"],"text":"basis-map coefficients of A026729 \u2192 D \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-74d14208bfebc28b","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-b7beb933369ebebf","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b7beb933369ebebf","other":{"id":"n-b7beb933369ebebf","name":"theta_plus_one_squared of theta_plus_one of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-8"],["0","0","54"]],"card_rows":[["1"],["0","-8"],["0","0","54"],["0","0","0","-384"]],"row_sums_preview":["1","-8","54","-384","3000","-25920","246960","-2580480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-19f259e2ece869e6","e-ace4307097e0c605"],"text":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-74d2bd1b03b14692","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-2d487bb06a750ce0","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-2d487bb06a750ce0","other":{"id":"n-2d487bb06a750ce0","name":"reversal of theta_plus_one of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6"],["18"]],"card_rows":[["1"],["6"],["18"],["40"]],"row_sums_preview":["1","6","18","40","75","126","196","288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["\u03b8+1","reverse"],"edges":["e-9b1f1f612dc2ed52","e-f8337255aed40935"],"text":"A127773 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A127773 \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-76a4dcb90b4c4e0f","kind":"fingerprint_related","node_a_id":"n-536ad043fc427a37","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-536ad043fc427a37","other":{"id":"n-536ad043fc427a37","name":"derivative of derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","-6"],["0","0","12"],["0","0","0","-20"]],"row_sums_preview":["0","0","2","-6","12","-20","30","-42"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["D","D"],"edges":["e-c22d2d4fa11fd8cc","e-a2617a4be5e1e6dc"],"text":"basis-map coefficients of A026729, shifted \u2192 D \u2192 D"},"headline":"basis-map coefficients of A026729, shifted \u2192 D \u2192 D","chain":null}},{"id":"l-7acc5bc4c39beecc","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-163dae1680588051","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-163dae1680588051","other":{"id":"n-163dae1680588051","name":"theta_plus_one of derivative of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["0","16"]],"card_rows":[["2"],["0","16"],["0","0","72"],["0","0","0","256"]],"row_sums_preview":["0","2","16","72","256","800","2304","6272"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D","\u03b8+1"],"edges":["e-6151a42dffed124f","e-4eda257bda86ad8d"],"text":"A130123 \u2192 D \u2192 \u03b8+1"},"headline":"A130123 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-7c679df8a2b6dde8","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-0a6318dfaf390653","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0a6318dfaf390653","other":{"id":"n-0a6318dfaf390653","name":"derivative of theta_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["0","42"]],"card_rows":[["8"],["0","42"],["0","0","120"],["0","0","0","260"]],"row_sums_preview":["0","8","42","120","260","480","798","1232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["\u03b8+1","D"],"edges":["e-5734274d793ba02f","e-0c9ac1cc78971525"],"text":"A131431 \u2192 \u03b8+1 \u2192 D"},"headline":"A131431 \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-7cdfde98a6be2308","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-4bc6a943f8bb5fc8","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-4bc6a943f8bb5fc8","other":{"id":"n-4bc6a943f8bb5fc8","name":"derivative of Triangle read by rows, (n-1) zeros followed by 2^n - 1.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["0","14"]],"card_rows":[["3"],["0","14"],["0","0","45"],["0","0","0","124"]],"row_sums_preview":["0","3","14","45","124","315","762","1785"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6394a556f7ad3cd4","steps":["D"],"edges":["e-fc7ee478d16b4e02"],"text":"A131436 \u2192 D"},"headline":"A131436 \u2192 D","chain":null}},{"id":"l-7d426fd3bd890ca6","kind":"fingerprint_related","node_a_id":"n-b8bb9d4926920605","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b8bb9d4926920605","other":{"id":"n-b8bb9d4926920605","name":"theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","2"],["0","0","6"]],"card_rows":[["1"],["0","2"],["0","0","6"],["0","0","0","24"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["\u03b8+1"],"edges":["e-4dc8adfab569fa5c"],"text":"A136572 \u2192 \u03b8+1"},"headline":"A136572 \u2192 \u03b8+1","chain":null}},{"id":"l-7eb870b09ef8ce57","kind":"fingerprint_related","node_a_id":"n-3cc7875cf81a5a84","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-3cc7875cf81a5a84","other":{"id":"n-3cc7875cf81a5a84","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-1"],["0","0","1"]],"card_rows":[["1"],["0","-1"],["0","0","1"],["0","0","0","-1"]],"row_sums_preview":["1","-1","1","-1","1","-1","1","-1"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":[],"edges":[],"text":"basis-map coefficients of A026729, shifted"},"headline":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","chain":null}},{"id":"l-7f8a14aa5ebf633d","kind":"fingerprint_related","node_a_id":"n-1bf99ad40f056d33","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-1bf99ad40f056d33","other":{"id":"n-1bf99ad40f056d33","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","0","1"]],"card_rows":[["1"],["0","1"],["0","0","1"],["0","0","0","1"]],"row_sums_preview":["1","1","1","1","1","1","1","1"],"regions":["[0, +inf)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":[],"edges":[],"text":"basis-map coefficients of A026729"},"headline":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","chain":null}},{"id":"l-7fbc5f142849f537","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-bc6d2dc805eb16d0","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-bc6d2dc805eb16d0","other":{"id":"n-bc6d2dc805eb16d0","name":"theta_plus_one_squared of hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","12"],["0","0","108"]],"card_rows":[["1"],["0","12"],["0","0","108"],["0","0","0","960"]],"row_sums_preview":["1","12","108","960","9000","90720","987840","11612160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f7eaad3557182df","steps":["(\u03b8+1)\u00b2"],"edges":["e-a7bc3318295c2a7e"],"text":"Hadamard-factor parent of A127773 \u2192 (\u03b8+1)\u00b2"},"headline":"Hadamard-factor parent of A127773 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-7fff56ed66b632ec","kind":"fingerprint_related","node_a_id":"n-7adc12e1650df214","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7adc12e1650df214","other":{"id":"n-7adc12e1650df214","name":"reversal of theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["3"]],"card_rows":[["1"],["2"],["3"],["4"]],"row_sums_preview":["1","2","3","4","5","6","7","8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["reverse"],"edges":["e-22381a2309b43121"],"text":"image under \u03b8+1 \u2192 reverse"},"headline":"image under \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-80a726574bb69e4f","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-0c63a514369dce6d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0c63a514369dce6d","other":{"id":"n-0c63a514369dce6d","name":"derivative of theta_plus_one of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["0","24"]],"card_rows":[["4"],["0","24"],["0","0","96"],["0","0","0","320"]],"row_sums_preview":["0","4","24","96","320","960","2688","7168"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["\u03b8+1","D"],"edges":["e-c327b1a68506ba1f","e-b51aab0b417706b2"],"text":"A130123 \u2192 \u03b8+1 \u2192 D"},"headline":"A130123 \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-813b5777c98e469a","kind":"fingerprint_related","node_a_id":"n-39194b37db596ae8","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-39194b37db596ae8","other":{"id":"n-39194b37db596ae8","name":"a(n) = 1 if n is a triangular number, otherwise 0.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1"]],"card_rows":[["1"],["1"],["1"],["1"]],"row_sums_preview":["1","1","1","1","1","1","1","1"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":["A010054"],"derivation":{"root":"n-39194b37db596ae8","steps":[],"edges":[],"text":"A010054"},"headline":"A010054: a(n) = 1 if n is a triangular number, otherwise 0.","chain":null}},{"id":"l-823d159d8ea3c243","kind":"fingerprint_related","node_a_id":"n-f479774b096b16a4","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-f479774b096b16a4","other":{"id":"n-f479774b096b16a4","name":"derivative of derivative of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","-6"],["0","0","12"],["0","0","0","-20"]],"row_sums_preview":["0","0","2","-6","12","-20","30","-42"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["D","D"],"edges":["e-7b1079618314be79","e-d0ab5723841dc1d1"],"text":"basis-map coefficients of A030528, shifted \u2192 D \u2192 D"},"headline":"basis-map coefficients of A030528, shifted \u2192 D \u2192 D","chain":null}},{"id":"l-83ad462021b66ee1","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-365fd9aa4da2786d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-365fd9aa4da2786d","other":{"id":"n-365fd9aa4da2786d","name":"reversal of theta_plus_one_squared of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16"],["63"]],"card_rows":[["1"],["16"],["63"],["160"]],"row_sums_preview":["1","16","63","160","325","576","931","1408"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-792e4f12e133c3d5","e-31adb5a7ab44e229"],"text":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-851eac65685a4bba","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-b23861b1e8ad6227","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b23861b1e8ad6227","other":{"id":"n-b23861b1e8ad6227","name":"reversal of theta_plus_one_squared of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4"],["9"]],"card_rows":[["1"],["-4"],["9"],["-16"]],"row_sums_preview":["1","-4","9","-16","25","-36","49","-64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-d58e809d3242b4fa","e-93eb7f27bdf514ad"],"text":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-86355898f86e078b","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-88aa9d1db9bf054a","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-88aa9d1db9bf054a","other":{"id":"n-88aa9d1db9bf054a","name":"theta_plus_one of hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","6"],["0","0","36"]],"card_rows":[["1"],["0","6"],["0","0","36"],["0","0","0","240"]],"row_sums_preview":["1","6","36","240","1800","15120","141120","1451520"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f7eaad3557182df","steps":["\u03b8+1"],"edges":["e-13a03af9f9756ffc"],"text":"Hadamard-factor parent of A127773 \u2192 \u03b8+1"},"headline":"Hadamard-factor parent of A127773 \u2192 \u03b8+1","chain":null}},{"id":"l-863cb521ab0579fd","kind":"fingerprint_related","node_a_id":"n-900a36a3001cf31d","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-900a36a3001cf31d","other":{"id":"n-900a36a3001cf31d","name":"derivative of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","4"]],"card_rows":[["1"],["0","4"],["0","0","18"],["0","0","0","96"]],"row_sums_preview":["0","1","4","18","96","600","4320","35280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["D"],"edges":["e-789a32df93c7cdb5"],"text":"A136572 \u2192 D"},"headline":"A136572 \u2192 D","chain":null}},{"id":"l-865989332f72884c","kind":"fingerprint_related","node_a_id":"n-de8575ce50ceeb5a","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-de8575ce50ceeb5a","other":{"id":"n-de8575ce50ceeb5a","name":"gamma_polynomial of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["6"]],"card_rows":[["1"],["2"],["6"],["20"]],"row_sums_preview":["1","2","6","20","70","252","924","3432"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["\u03b3"],"edges":["e-7d7c781fa7a59405"],"text":"A356546 \u2192 \u03b3"},"headline":"A356546 \u2192 \u03b3","chain":null}},{"id":"l-875c89e0afce197c","kind":"fingerprint_related","node_a_id":"n-e3a53e103103e79e","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e3a53e103103e79e","other":{"id":"n-e3a53e103103e79e","name":"theta_plus_one_squared of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","12"],["0","0","54"]],"card_rows":[["1"],["0","12"],["0","0","54"],["0","0","0","160"]],"row_sums_preview":["1","12","54","160","375","756","1372","2304"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["(\u03b8+1)\u00b2"],"edges":["e-e78403b3b87fde04"],"text":"A127773 \u2192 (\u03b8+1)\u00b2"},"headline":"A127773 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-87ee9c001d0277d1","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-ed990bb2b2d5cad1","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ed990bb2b2d5cad1","other":{"id":"n-ed990bb2b2d5cad1","name":"theta_plus_one_squared of derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["0","8"],["0","0","0","64"]],"card_rows":[["0","8"],["0","0","0","64"],["0","0","0","0","0","216"],["0","0","0","0","0","0","0","512"]],"row_sums_preview":["0","8","64","216","512","1000","1728","2744"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-521a2a476289c028","e-76ad704f95d62f3d"],"text":"basis-map coefficients of A297191, shifted \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A297191, shifted \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-88127148885fa899","kind":"fingerprint_related","node_a_id":"n-7b3743413970d91c","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7b3743413970d91c","other":{"id":"n-7b3743413970d91c","name":"theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","2"],["0","0","3"]],"card_rows":[["1"],["0","2"],["0","0","3"],["0","0","0","4"]],"row_sums_preview":["1","2","3","4","5","6","7","8"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":[],"edges":[],"text":"image under \u03b8+1"},"headline":"theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","chain":null}},{"id":"l-882de591d9798c75","kind":"fingerprint_related","node_a_id":"n-a3e72bc3107566a5","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-a3e72bc3107566a5","other":{"id":"n-a3e72bc3107566a5","name":"derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","6"],["0","0","12"],["0","0","0","20"]],"row_sums_preview":["0","0","2","6","12","20","30","42"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["D"],"edges":["e-1aec373ef809ab0b"],"text":"A127648 \u2192 D"},"headline":"A127648 \u2192 D","chain":null}},{"id":"l-8830bd354dfc67c2","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-13d0b717287e58a3","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-13d0b717287e58a3","other":{"id":"n-13d0b717287e58a3","name":"derivative of derivative of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["4"]],"card_rows":[["4"],["0","36"],["0","0","288"],["0","0","0","2400"]],"row_sums_preview":["0","0","4","36","288","2400","21600","211680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["D","D"],"edges":["e-789a32df93c7cdb5","e-1d42a808c9ebd63b"],"text":"A136572 \u2192 D \u2192 D"},"headline":"A136572 \u2192 D \u2192 D","chain":null}},{"id":"l-8a15a36ee1b42ebe","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-81c4d9ad43751521","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-81c4d9ad43751521","other":{"id":"n-81c4d9ad43751521","name":"theta_plus_one of Triangle read by rows, (n-1) zeros followed by 2^n - 1.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","6"],["0","0","21"]],"card_rows":[["1"],["0","6"],["0","0","21"],["0","0","0","60"]],"row_sums_preview":["1","6","21","60","155","378","889","2040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6394a556f7ad3cd4","steps":["\u03b8+1"],"edges":["e-8378fa1ebe28167a"],"text":"A131436 \u2192 \u03b8+1"},"headline":"A131436 \u2192 \u03b8+1","chain":null}},{"id":"l-8a908a6db46c3fe9","kind":"fingerprint_related","node_a_id":"n-bf79b715e07bc7a5","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-bf79b715e07bc7a5","other":{"id":"n-bf79b715e07bc7a5","name":"Triangle read by rows: T(m,n) = parity of 0^n + 0^m, n = 0,1,2,3 ..., m = 0,1,2,3, ... n.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["1"]],"card_rows":[["1"],["1"],["1"],["1"]],"row_sums_preview":["0","1","1","1","1","1","1","1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A073424"],"derivation":{"root":"n-bf79b715e07bc7a5","steps":[],"edges":[],"text":"A073424"},"headline":"A073424: Triangle read by rows: T(m,n) = parity of 0^n + 0^m, n = 0,1,2,3 ..., m = 0,1,2,3, ... n.","chain":null}},{"id":"l-8c5e8b9e5068312e","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-4a40ba7b502feaf2","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-4a40ba7b502feaf2","other":{"id":"n-4a40ba7b502feaf2","name":"theta_plus_one_squared of theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","8"],["0","0","54"]],"card_rows":[["1"],["0","8"],["0","0","54"],["0","0","0","384"]],"row_sums_preview":["1","8","54","384","3000","25920","246960","2580480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-76872140ee232c50","e-2bd2515cd2bf4c20"],"text":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-8e3a8a67bb012adf","kind":"fingerprint_related","node_a_id":"n-6394a556f7ad3cd4","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-6394a556f7ad3cd4","other":{"id":"n-6394a556f7ad3cd4","name":"Triangle read by rows, (n-1) zeros followed by 2^n - 1.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","3"],["0","0","7"]],"card_rows":[["1"],["0","3"],["0","0","7"],["0","0","0","15"]],"row_sums_preview":["1","3","7","15","31","63","127","255"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A131436"],"derivation":{"root":"n-6394a556f7ad3cd4","steps":[],"edges":[],"text":"A131436"},"headline":"A131436: Triangle read by rows, (n-1) zeros followed by 2^n - 1.","chain":null}},{"id":"l-9268fbd32c0a7ad2","kind":"fingerprint_related","node_a_id":"n-7f02402e8837dcc8","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7f02402e8837dcc8","other":{"id":"n-7f02402e8837dcc8","name":"theta_plus_one_squared of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","8"],["0","0","36"]],"card_rows":[["1"],["0","8"],["0","0","36"],["0","0","0","128"]],"row_sums_preview":["1","8","36","128","400","1152","3136","8192"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["(\u03b8+1)\u00b2"],"edges":["e-c89dffe6ace51b80"],"text":"A130123 \u2192 (\u03b8+1)\u00b2"},"headline":"A130123 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-93c4a259b1e098f5","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-3f6bfa307c4719f3","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-3f6bfa307c4719f3","other":{"id":"n-3f6bfa307c4719f3","name":"derivative of derivative of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],[]],"card_rows":[["6"],["0","24"],["0","0","60"],["0","0","0","120"]],"row_sums_preview":["0","0","0","6","24","60","120","210"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["D","D"],"edges":["e-1aec373ef809ab0b","e-3f14a0a1652e5249"],"text":"A127648 \u2192 D \u2192 D"},"headline":"A127648 \u2192 D \u2192 D","chain":null}},{"id":"l-9404252099299e46","kind":"fingerprint_related","node_a_id":"n-38e0e1d209c2aec5","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-38e0e1d209c2aec5","other":{"id":"n-38e0e1d209c2aec5","name":"theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-4"],["0","0","9"]],"card_rows":[["1"],["0","-4"],["0","0","9"],["0","0","0","-16"]],"row_sums_preview":["1","-4","9","-16","25","-36","49","-64"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["(\u03b8+1)\u00b2"],"edges":["e-23b8fb76a1e0b114"],"text":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-9479692b5c788ac4","kind":"fingerprint_related","node_a_id":"n-900171d522524464","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-900171d522524464","other":{"id":"n-900171d522524464","name":"theta_plus_one of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","6"],["0","0","18"]],"card_rows":[["1"],["0","6"],["0","0","18"],["0","0","0","40"]],"row_sums_preview":["1","6","18","40","75","126","196","288"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["\u03b8+1"],"edges":["e-9b1f1f612dc2ed52"],"text":"A127773 \u2192 \u03b8+1"},"headline":"A127773 \u2192 \u03b8+1","chain":null}},{"id":"l-96637cd091017002","kind":"fingerprint_related","node_a_id":"n-a895a528c93c18ea","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-a895a528c93c18ea","other":{"id":"n-a895a528c93c18ea","name":"Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","0","2"]],"card_rows":[["1"],["0","1"],["0","0","2"],["0","0","0","4"]],"row_sums_preview":["1","1","2","4","8","16","32","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A134309"],"derivation":{"root":"n-a895a528c93c18ea","steps":[],"edges":[],"text":"A134309"},"headline":"A134309: Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).","chain":null}},{"id":"l-97072b432a6a48e7","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-72f808c7c4297337","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-72f808c7c4297337","other":{"id":"n-72f808c7c4297337","name":"theta_plus_one_squared of Triangle read by rows, (n-1) zeros followed by 2^n - 1.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","12"],["0","0","63"]],"card_rows":[["1"],["0","12"],["0","0","63"],["0","0","0","240"]],"row_sums_preview":["1","12","63","240","775","2268","6223","16320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6394a556f7ad3cd4","steps":["(\u03b8+1)\u00b2"],"edges":["e-b067f446be477ee1"],"text":"A131436 \u2192 (\u03b8+1)\u00b2"},"headline":"A131436 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-983b4100ac38c252","kind":"fingerprint_related","node_a_id":"n-de52277bee139759","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-de52277bee139759","other":{"id":"n-de52277bee139759","name":"derivative of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["0","2"]],"card_rows":[["-1"],["0","2"],["0","0","-3"],["0","0","0","4"]],"row_sums_preview":["0","-1","2","-3","4","-5","6","-7"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["D"],"edges":["e-7b1079618314be79"],"text":"basis-map coefficients of A030528, shifted \u2192 D"},"headline":"basis-map coefficients of A030528, shifted \u2192 D","chain":null}},{"id":"l-9c4b5cf9a18940b6","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-d8d329506f23ca59","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-d8d329506f23ca59","other":{"id":"n-d8d329506f23ca59","name":"theta_plus_one_squared of theta_plus_one of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","16"]],"card_rows":[["1"],["0","16"],["0","0","81"],["0","0","0","256"]],"row_sums_preview":["0","1","16","81","256","625","1296","2401"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-0695c41826b28137","e-408dd39129505bdb"],"text":"A132440 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A132440 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-9d14541247f9ff2b","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-7f6e7eeee974a80c","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7f6e7eeee974a80c","other":{"id":"n-7f6e7eeee974a80c","name":"theta_plus_one_squared of theta_plus_one of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","16"],["0","0","108"]],"card_rows":[["1"],["0","16"],["0","0","108"],["0","0","0","512"]],"row_sums_preview":["1","16","108","512","2000","6912","21952","65536"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-c89dffe6ace51b80","e-144614e63e02c9f0"],"text":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-9e7cc123775ce7bf","kind":"fingerprint_related","node_a_id":"n-a23fdacbe8c6317d","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-a23fdacbe8c6317d","other":{"id":"n-a23fdacbe8c6317d","name":"derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["0","18"]],"card_rows":[["4"],["0","18"],["0","0","48"],["0","0","0","100"]],"row_sums_preview":["0","4","18","48","100","180","294","448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D"],"edges":["e-06738ea5bb413735","e-864c354a3903aafb"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-9fcbf5e628009ade","kind":"fingerprint_related","node_a_id":"n-0efca9aef0911045","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0efca9aef0911045","other":{"id":"n-0efca9aef0911045","name":"veronese_section_2_1 of A natural number operator.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["4"]],"card_rows":[["3"],["4"],["0","7"],["0","8"]],"row_sums_preview":["0","3","4","7","8","11","12","15"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f4c1dee76f5a68b2","steps":["Veronese 2 (j=1)"],"edges":["e-611716aca3d048f2"],"text":"A132774 \u2192 Veronese 2 (j=1)"},"headline":"A132774 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-a5b7a644a5ac5b2c","kind":"fingerprint_related","node_a_id":"n-1c11ffc3c4cd47ed","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-1c11ffc3c4cd47ed","other":{"id":"n-1c11ffc3c4cd47ed","name":"Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","2"]],"card_rows":[["1"],["0","2"],["0","0","3"],["0","0","0","4"]],"row_sums_preview":["0","1","2","3","4","5","6","7"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A132440"],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":[],"edges":[],"text":"A132440"},"headline":"A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","chain":null}},{"id":"l-a79be6c45f13110d","kind":"fingerprint_related","node_a_id":"n-d6ec17c5486fb722","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-d6ec17c5486fb722","other":{"id":"n-d6ec17c5486fb722","name":"Triangle read by rows: row n consists of n zeros followed by n+1.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","2"],["0","0","3"]],"card_rows":[["1"],["0","2"],["0","0","3"],["0","0","0","4"]],"row_sums_preview":["1","2","3","4","5","6","7","8"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A127648"],"derivation":{"root":"n-d6ec17c5486fb722","steps":[],"edges":[],"text":"A127648"},"headline":"A127648: Triangle read by rows: row n consists of n zeros followed by n+1.","chain":null}},{"id":"l-a95bb43e3a22a106","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-b976c6f522a327a2","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b976c6f522a327a2","other":{"id":"n-b976c6f522a327a2","name":"theta_plus_one_squared of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-16"],["0","0","81"]],"card_rows":[["1"],["0","-16"],["0","0","81"],["0","0","0","-256"]],"row_sums_preview":["1","-16","81","-256","625","-1296","2401","-4096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-23b8fb76a1e0b114","e-4541cc1254718e35"],"text":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-aa72c11d8d8468c7","kind":"fingerprint_related","node_a_id":"n-d09dd8597f391f42","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-d09dd8597f391f42","other":{"id":"n-d09dd8597f391f42","name":"gamma_polynomial of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["9"]],"card_rows":[["1"],["3"],["9"],["27"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["\u03b3"],"edges":["e-b8c83da274fc75bd"],"text":"A038221 \u2192 \u03b3"},"headline":"A038221 \u2192 \u03b3","chain":null}},{"id":"l-b16231ad62c37f88","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-38d69968715621b9","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-38d69968715621b9","other":{"id":"n-38d69968715621b9","name":"derivative of theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["0","36"]],"card_rows":[["4"],["0","36"],["0","0","288"],["0","0","0","2400"]],"row_sums_preview":["0","4","36","288","2400","21600","211680","2257920"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-76872140ee232c50","e-db679486a067259c"],"text":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-b1d35fd4285fb5f9","kind":"fingerprint_related","node_a_id":"n-0d90c938f81b41a4","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0d90c938f81b41a4","other":{"id":"n-0d90c938f81b41a4","name":"gamma_polynomial of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5"],["15"]],"card_rows":[["1"],["5"],["15"],["35"]],"row_sums_preview":["1","5","15","35","70","126","210","330"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["\u03b3"],"edges":["e-70e9e46985bbe802"],"text":"A178821 \u2192 \u03b3"},"headline":"A178821 \u2192 \u03b3","chain":null}},{"id":"l-b21297b639f81a70","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-6fcd40d030ea78a2","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-6fcd40d030ea78a2","other":{"id":"n-6fcd40d030ea78a2","name":"derivative of theta_plus_one_squared of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["16"],["0","126"]],"card_rows":[["16"],["0","126"],["0","0","480"],["0","0","0","1300"]],"row_sums_preview":["0","16","126","480","1300","2880","5586","9856"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-792e4f12e133c3d5","e-3dcbf9c492ee84ae"],"text":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-b22650d6cae7e2b6","kind":"fingerprint_related","node_a_id":"n-0e16e206534c379c","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-0e16e206534c379c","other":{"id":"n-0e16e206534c379c","name":"veronese_section_2_0 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["0","1"]],"card_rows":[["1"],["1"],["0","1"],["0","1"]],"row_sums_preview":["1","1","1","1","1","1","1","1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea2139b8dbf6bf8d","steps":["Veronese 2"],"edges":["e-85a0cad9cd09d768"],"text":"A133080 \u2192 Veronese 2"},"headline":"A133080 \u2192 Veronese 2","chain":null}},{"id":"l-b2b860005d5d9e05","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-c562221857c4b472","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-c562221857c4b472","other":{"id":"n-c562221857c4b472","name":"derivative of theta_plus_one of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["0","54"]],"card_rows":[["8"],["0","54"],["0","0","192"],["0","0","0","500"]],"row_sums_preview":["0","8","54","192","500","1080","2058","3584"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-048aefa516697baf","e-3f6bb02a2094ad86"],"text":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-b556b11f53ff15b2","kind":"fingerprint_related","node_a_id":"n-d51e4f59f14ff00f","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-d51e4f59f14ff00f","other":{"id":"n-d51e4f59f14ff00f","name":"Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","4"],["0","0","9"]],"card_rows":[["1"],["0","4"],["0","0","9"],["0","0","0","16"]],"row_sums_preview":["1","4","9","16","25","36","49","64"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A127733"],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1"],"edges":["e-06738ea5bb413735"],"text":"image under \u03b8+1 \u2192 \u03b8+1"},"headline":"A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","chain":"image under \u03b8+1 \u2192 \u03b8+1"}},{"id":"l-b5cf2a7fe87ab766","kind":"fingerprint_related","node_a_id":"n-ae7910bbe46f8a41","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ae7910bbe46f8a41","other":{"id":"n-ae7910bbe46f8a41","name":"Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","2"],["0","0","4"]],"card_rows":[["1"],["0","2"],["0","0","4"],["0","0","0","8"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A130123"],"derivation":{"root":"n-ae7910bbe46f8a41","steps":[],"edges":[],"text":"A130123"},"headline":"A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","chain":null}},{"id":"l-b5d191448ca0952b","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-25da99b3d48410b5","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-25da99b3d48410b5","other":{"id":"n-25da99b3d48410b5","name":"derivative of theta_plus_one_squared of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","18"],["0","0","48"],["0","0","0","100"]],"card_rows":[["0","18"],["0","0","48"],["0","0","0","100"],["0","0","0","0","180"]],"row_sums_preview":["18","48","100","180","294","448","648","900"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-c527a11d264443d6","e-9a294a9b2408f2b9"],"text":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-b807e4adf5df68eb","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-361bc98d6a30cb1a","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-361bc98d6a30cb1a","other":{"id":"n-361bc98d6a30cb1a","name":"derivative of Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","4"]],"card_rows":[["1"],["0","4"],["0","0","12"],["0","0","0","32"]],"row_sums_preview":["0","1","4","12","32","80","192","448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a895a528c93c18ea","steps":["D"],"edges":["e-ec7103578cbb2e54"],"text":"A134309 \u2192 D"},"headline":"A134309 \u2192 D","chain":null}},{"id":"l-b8995b938f51a08f","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-ef108e007c0ea024","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ef108e007c0ea024","other":{"id":"n-ef108e007c0ea024","name":"derivative of derivative of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["8"]],"card_rows":[["8"],["0","48"],["0","0","192"],["0","0","0","640"]],"row_sums_preview":["0","0","8","48","192","640","1920","5376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D","D"],"edges":["e-6151a42dffed124f","e-d6dbfe3014fc3205"],"text":"A130123 \u2192 D \u2192 D"},"headline":"A130123 \u2192 D \u2192 D","chain":null}},{"id":"l-b9abac8479247433","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-17b876afd76196ce","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-17b876afd76196ce","other":{"id":"n-17b876afd76196ce","name":"reversal of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4"],["9"]],"card_rows":[["1"],["4"],["9"],["16"]],"row_sums_preview":["1","4","9","16","25","36","49","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-3411eba2e020f0d3","e-1214990321c565b3"],"text":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-b9b29dd72dad3b99","kind":"fingerprint_related","node_a_id":"n-477d9807e1cbb785","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-477d9807e1cbb785","other":{"id":"n-477d9807e1cbb785","name":"basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-1"],["0","0","1"]],"card_rows":[["1"],["0","-1"],["0","0","1"],["0","0","0","-1"]],"row_sums_preview":["1","-1","1","-1","1","-1","1","-1"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":[],"edges":[],"text":"basis-map coefficients of A030528, shifted"},"headline":"basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","chain":null}},{"id":"l-b9cb8afcf39df9db","kind":"fingerprint_related","node_a_id":"n-fbb4b9430d9de872","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-fbb4b9430d9de872","other":{"id":"n-fbb4b9430d9de872","name":"Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","0","2"]],"card_rows":[["1"],["0","1"],["0","0","2"],["0","0","0","6"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A136572"],"derivation":{"root":"n-fbb4b9430d9de872","steps":[],"edges":[],"text":"A136572"},"headline":"A136572: Triangle read by rows: row n consists of n zeros followed by n!.","chain":null}},{"id":"l-ba394ffd0df55214","kind":"fingerprint_related","node_a_id":"n-6299862621ff8810","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-6299862621ff8810","other":{"id":"n-6299862621ff8810","name":"Expansion of the rotational partition function for a heteronuclear diatomic molecule.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["5"]],"card_rows":[["1"],["3"],["5"],["7"]],"row_sums_preview":["1","3","5","7","9","11","13","15"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","(-inf, 0)","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":["A198954"],"derivation":{"root":"n-6299862621ff8810","steps":[],"edges":[],"text":"A198954"},"headline":"A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.","chain":null}},{"id":"l-bb1149220eac060b","kind":"fingerprint_related","node_a_id":"n-9b81dfafd0e0e0fd","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-9b81dfafd0e0e0fd","other":{"id":"n-9b81dfafd0e0e0fd","name":"Shift operator, right.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","1"]],"card_rows":[["1"],["0","1"],["0","0","1"],["0","0","0","1"]],"row_sums_preview":["0","1","1","1","1","1","1","1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A129184"],"derivation":{"root":"n-9b81dfafd0e0e0fd","steps":[],"edges":[],"text":"A129184"},"headline":"A129184: Shift operator, right.","chain":null}},{"id":"l-bc79d82eb9baed2a","kind":"fingerprint_related","node_a_id":"n-d2ea2ff0a779ba6e","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-d2ea2ff0a779ba6e","other":{"id":"n-d2ea2ff0a779ba6e","name":"Right shift operator generating 1's in shifted spaces.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"irregular","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["0","1"]],"card_rows":[["1"],["1"],["0","1"],["0","0","1"]],"row_sums_preview":["1","1","1","1","1","1","1","1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A129186"],"derivation":{"root":"n-d2ea2ff0a779ba6e","steps":[],"edges":[],"text":"A129186"},"headline":"A129186: Right shift operator generating 1's in shifted spaces.","chain":null}},{"id":"l-bcb8e550eff9666e","kind":"fingerprint_related","node_a_id":"n-1f7eaad3557182df","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-1f7eaad3557182df","other":{"id":"n-1f7eaad3557182df","name":"hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","3"],["0","0","12"]],"card_rows":[["1"],["0","3"],["0","0","12"],["0","0","0","60"]],"row_sums_preview":["1","3","12","60","360","2520","20160","181440"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f7eaad3557182df","steps":[],"edges":[],"text":"Hadamard-factor parent of A127773"},"headline":"hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","chain":null}},{"id":"l-bea675160317da95","kind":"fingerprint_related","node_a_id":"n-127551b414704b9e","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-127551b414704b9e","other":{"id":"n-127551b414704b9e","name":"gamma_polynomial of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["2"]],"card_rows":[["1"],["2"],["2"],["4"]],"row_sums_preview":["1","2","2","4","4","8","8","16"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["\u03b3"],"edges":["e-71d067081fff0c94"],"text":"A135838 \u2192 \u03b3"},"headline":"A135838 \u2192 \u03b3","chain":null}},{"id":"l-bef5160b58cfc654","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-73608580bc5459cb","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-73608580bc5459cb","other":{"id":"n-73608580bc5459cb","name":"theta_plus_one_squared of theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","0","27"],["0","0","0","0","125"]],"card_rows":[["1"],["0","0","27"],["0","0","0","0","125"],["0","0","0","0","0","0","343"]],"row_sums_preview":["1","27","125","343","729","1331","2197","3375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-5ac7ff393747d6fc","e-6193a8ef14737117"],"text":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-c0bdd8b0437b8425","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-8a7c684177214e25","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-8a7c684177214e25","other":{"id":"n-8a7c684177214e25","name":"theta_plus_one of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","8"],["0","0","27"]],"card_rows":[["1"],["0","8"],["0","0","27"],["0","0","0","64"]],"row_sums_preview":["1","8","27","64","125","216","343","512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-3411eba2e020f0d3","e-34d310309be943ad"],"text":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-c0da4165310fed53","kind":"fingerprint_related","node_a_id":"n-efb24c96253c2b9d","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-efb24c96253c2b9d","other":{"id":"n-efb24c96253c2b9d","name":"theta_plus_one_squared of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","9"],["0","0","0","16"],["0","0","0","0","25"]],"card_rows":[["0","0","9"],["0","0","0","16"],["0","0","0","0","25"],["0","0","0","0","0","36"]],"row_sums_preview":["9","16","25","36","49","64","81","100"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["(\u03b8+1)\u00b2"],"edges":["e-c527a11d264443d6"],"text":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-c18a1289d3b7bac3","kind":"fingerprint_related","node_a_id":"n-eadc59843e79d6ec","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-eadc59843e79d6ec","other":{"id":"n-eadc59843e79d6ec","name":"theta_plus_one of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","4"],["0","0","12"]],"card_rows":[["1"],["0","4"],["0","0","12"],["0","0","0","32"]],"row_sums_preview":["1","4","12","32","80","192","448","1024"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["\u03b8+1"],"edges":["e-c327b1a68506ba1f"],"text":"A130123 \u2192 \u03b8+1"},"headline":"A130123 \u2192 \u03b8+1","chain":null}},{"id":"l-c1ccc4a7b64daa5a","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-fa5cb12d3c0ddf59","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-fa5cb12d3c0ddf59","other":{"id":"n-fa5cb12d3c0ddf59","name":"derivative of theta_plus_one_squared of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["0","72"]],"card_rows":[["8"],["0","72"],["0","0","384"],["0","0","0","1600"]],"row_sums_preview":["0","8","72","384","1600","5760","18816","57344"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-c89dffe6ace51b80","e-a41c53594a713db1"],"text":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-c32a859c34b12951","kind":"fingerprint_related","node_a_id":"n-b186435e005aa2fb","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-b186435e005aa2fb","other":{"id":"n-b186435e005aa2fb","name":"theta_plus_one_squared of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-4"],["0","0","9"]],"card_rows":[["1"],["0","-4"],["0","0","9"],["0","0","0","-16"]],"row_sums_preview":["1","-4","9","-16","25","-36","49","-64"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["(\u03b8+1)\u00b2"],"edges":["e-d58e809d3242b4fa"],"text":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-c41a8602f8000e2b","kind":"fingerprint_related","node_a_id":"n-511a2225fa2191c9","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-511a2225fa2191c9","other":{"id":"n-511a2225fa2191c9","name":"Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","3"],["0","0","6"]],"card_rows":[["1"],["0","3"],["0","0","6"],["0","0","0","10"]],"row_sums_preview":["1","3","6","10","15","21","28","36"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A127773"],"derivation":{"root":"n-511a2225fa2191c9","steps":[],"edges":[],"text":"A127773"},"headline":"A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","chain":null}},{"id":"l-c6a4383cd5fd55c0","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-eee2645ab36c5341","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-eee2645ab36c5341","other":{"id":"n-eee2645ab36c5341","name":"theta_plus_one_squared of theta_plus_one_squared of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","32"],["0","0","324"]],"card_rows":[["1"],["0","32"],["0","0","324"],["0","0","0","2048"]],"row_sums_preview":["1","32","324","2048","10000","41472","153664","524288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-c89dffe6ace51b80","e-627ed621ce0265d9"],"text":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-c6d8f54b5651a58c","kind":"fingerprint_related","node_a_id":"n-101e6570094d0688","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-101e6570094d0688","other":{"id":"n-101e6570094d0688","name":"gamma_polynomial of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144"],["1728"],["20736"]],"card_rows":[["144"],["1728"],["20736"],["248832"]],"row_sums_preview":["144","1728","20736","248832","2985984","35831808","429981696","5159780352"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["\u03b3"],"edges":["e-5535504a50c93560"],"text":"A038338 \u2192 \u03b3"},"headline":"A038338 \u2192 \u03b3","chain":null}},{"id":"l-c75204d1f1106d2e","kind":"fingerprint_related","node_a_id":"n-2cec638375170207","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-2cec638375170207","other":{"id":"n-2cec638375170207","name":"theta_plus_one_squared of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-4"],["0","0","18"]],"card_rows":[["1"],["0","-4"],["0","0","18"],["0","0","0","-96"]],"row_sums_preview":["1","-4","18","-96","600","-4320","35280","-322560"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["(\u03b8+1)\u00b2"],"edges":["e-3d038fb38f7286b3"],"text":"basis-map coefficients of A021012 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A021012 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-c7ac03ee6e418fea","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-045d8149da50c956","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-045d8149da50c956","other":{"id":"n-045d8149da50c956","name":"theta_plus_one of derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["0","4"],["0","0","0","16"]],"card_rows":[["0","4"],["0","0","0","16"],["0","0","0","0","0","36"],["0","0","0","0","0","0","0","64"]],"row_sums_preview":["0","4","16","36","64","100","144","196"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["D","\u03b8+1"],"edges":["e-521a2a476289c028","e-8a0b6bd8d534bc6f"],"text":"basis-map coefficients of A297191, shifted \u2192 D \u2192 \u03b8+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-c8f53853b85edf01","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-ea47b3ca22306aaf","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ea47b3ca22306aaf","other":{"id":"n-ea47b3ca22306aaf","name":"derivative of derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["0","0","12"]],"card_rows":[["2"],["0","0","12"],["0","0","0","0","30"],["0","0","0","0","0","0","56"]],"row_sums_preview":["0","2","12","30","56","90","132","182"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["D","D"],"edges":["e-521a2a476289c028","e-43c7e5b9695d19e3"],"text":"basis-map coefficients of A297191, shifted \u2192 D \u2192 D"},"headline":"basis-map coefficients of A297191, shifted \u2192 D \u2192 D","chain":null}},{"id":"l-c96d22523aa37bc6","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-44a466625b8ccafc","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-44a466625b8ccafc","other":{"id":"n-44a466625b8ccafc","name":"reversal of theta_plus_one of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4"],["12"]],"card_rows":[["1"],["4"],["12"],["32"]],"row_sums_preview":["1","4","12","32","80","192","448","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["\u03b8+1","reverse"],"edges":["e-c327b1a68506ba1f","e-c9d6e2b1b73558ff"],"text":"A130123 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A130123 \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-cd18a04e6c0396dc","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-5f818df3c568d1ec","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-5f818df3c568d1ec","other":{"id":"n-5f818df3c568d1ec","name":"reversal of theta_plus_one of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8"],["27"]],"card_rows":[["1"],["8"],["27"],["64"]],"row_sums_preview":["1","8","27","64","125","216","343","512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-048aefa516697baf","e-15f247657cb30da8"],"text":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-d140a9ec5dcce583","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-591f343a6759e331","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-591f343a6759e331","other":{"id":"n-591f343a6759e331","name":"theta_plus_one_squared of theta_plus_one_squared of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-16"],["0","0","81"]],"card_rows":[["1"],["0","-16"],["0","0","81"],["0","0","0","-256"]],"row_sums_preview":["1","-16","81","-256","625","-1296","2401","-4096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-d58e809d3242b4fa","e-8011541b0d3d5108"],"text":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-d1576a4187e0c08a","kind":"fingerprint_related","node_a_id":"n-cdb0db4ad473ee28","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-cdb0db4ad473ee28","other":{"id":"n-cdb0db4ad473ee28","name":"basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","0","1"],["0","0","0","0","1"]],"card_rows":[["1"],["0","0","1"],["0","0","0","0","1"],["0","0","0","0","0","0","1"]],"row_sums_preview":["1","1","1","1","1","1","1","1"],"regions":["(-inf, 0]","[0, +inf)","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":[],"edges":[],"text":"basis-map coefficients of A297191, shifted"},"headline":"basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","chain":null}},{"id":"l-d5ee2593816f77da","kind":"fingerprint_related","node_a_id":"n-7a57e4a2d6893547","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-7a57e4a2d6893547","other":{"id":"n-7a57e4a2d6893547","name":"derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["0","14"]],"card_rows":[["4"],["0","14"],["0","0","30"],["0","0","0","52"]],"row_sums_preview":["0","4","14","30","52","80","114","154"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D"],"edges":["e-7e9b074076a3cdbe"],"text":"A131431 \u2192 D"},"headline":"A131431 \u2192 D","chain":null}},{"id":"l-d9f918ff3ad614c1","kind":"fingerprint_related","node_a_id":"n-89bc7751aef505c2","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-89bc7751aef505c2","other":{"id":"n-89bc7751aef505c2","name":"theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","4"],["0","0","9"]],"card_rows":[["1"],["0","4"],["0","0","9"],["0","0","0","16"]],"row_sums_preview":["1","4","9","16","25","36","49","64"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["(\u03b8+1)\u00b2"],"edges":["e-3411eba2e020f0d3"],"text":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-da3496769ef3b152","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-f800fff69bd7f855","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-f800fff69bd7f855","other":{"id":"n-f800fff69bd7f855","name":"theta_plus_one of theta_plus_one_squared of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","32"],["0","0","189"]],"card_rows":[["1"],["0","32"],["0","0","189"],["0","0","0","640"]],"row_sums_preview":["1","32","189","640","1625","3456","6517","11264"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-792e4f12e133c3d5","e-00c0bc35c7e33bbf"],"text":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-db00f3b3608071b4","kind":"fingerprint_related","node_a_id":"n-e8ea751e7d01e0f2","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e8ea751e7d01e0f2","other":{"id":"n-e8ea751e7d01e0f2","name":"reversal of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4"],["9"]],"card_rows":[["1"],["4"],["9"],["16"]],"row_sums_preview":["1","4","9","16","25","36","49","64"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","reverse"],"edges":["e-06738ea5bb413735","e-012d45056c0bdaeb"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 reverse"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-dca05fd3aa9b33f2","kind":"fingerprint_related","node_a_id":"n-1ecb953a76cff627","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-1ecb953a76cff627","other":{"id":"n-1ecb953a76cff627","name":"Sums of two powers of zero: triangle read by rows: T(m,n) = 0^n + 0^m, n >= 0, m = 0..n.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["1"],["1"]],"card_rows":[["2"],["1"],["1"],["1"]],"row_sums_preview":["2","1","1","1","1","1","1","1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A073423"],"derivation":{"root":"n-1ecb953a76cff627","steps":[],"edges":[],"text":"A073423"},"headline":"A073423: Sums of two powers of zero: triangle read by rows: T(m,n) = 0^n + 0^m, n >= 0, m = 0..n.","chain":null}},{"id":"l-dcd3f2c507bde7c0","kind":"fingerprint_related","node_a_id":"n-5fe240122e98ffb3","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-5fe240122e98ffb3","other":{"id":"n-5fe240122e98ffb3","name":"basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","1"],["0","0","0","1"],["0","0","0","0","1"]],"card_rows":[["0","0","1"],["0","0","0","1"],["0","0","0","0","1"],["0","0","0","0","0","1"]],"row_sums_preview":["1","1","1","1","1","1","1","1"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":[],"edges":[],"text":"basis-map coefficients of A112227"},"headline":"basis-map coefficients of A scaled Hermite triangle.","chain":null}},{"id":"l-dcec8705f0c06a71","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-bc00eb3b372f5b29","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-bc00eb3b372f5b29","other":{"id":"n-bc00eb3b372f5b29","name":"derivative of hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["0","24"]],"card_rows":[["3"],["0","24"],["0","0","180"],["0","0","0","1440"]],"row_sums_preview":["0","3","24","180","1440","12600","120960","1270080"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f7eaad3557182df","steps":["D"],"edges":["e-68c35a590d8496af"],"text":"Hadamard-factor parent of A127773 \u2192 D"},"headline":"Hadamard-factor parent of A127773 \u2192 D","chain":null}},{"id":"l-de1e28ece8fb3868","kind":"fingerprint_related","node_a_id":"n-555e97d767a25a49","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-555e97d767a25a49","other":{"id":"n-555e97d767a25a49","name":"derivative of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["0","8"]],"card_rows":[["2"],["0","8"],["0","0","24"],["0","0","0","64"]],"row_sums_preview":["0","2","8","24","64","160","384","896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D"],"edges":["e-6151a42dffed124f"],"text":"A130123 \u2192 D"},"headline":"A130123 \u2192 D","chain":null}},{"id":"l-de6177a78b97a3ee","kind":"fingerprint_related","node_a_id":"n-2d329d45d9776260","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-2d329d45d9776260","other":{"id":"n-2d329d45d9776260","name":"theta_plus_one_squared of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","16"],["0","0","81"]],"card_rows":[["1"],["0","16"],["0","0","81"],["0","0","0","256"]],"row_sums_preview":["1","16","81","256","625","1296","2401","4096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-06738ea5bb413735","e-7bd1af475453dbed"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-deb10a7042dc70e1","kind":"fingerprint_related","node_a_id":"n-e9cf5b3881297005","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e9cf5b3881297005","other":{"id":"n-e9cf5b3881297005","name":"derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","2"]],"card_rows":[["1"],["0","2"],["0","0","3"],["0","0","0","4"]],"row_sums_preview":["0","1","2","3","4","5","6","7"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["D"],"edges":["e-198d139196a8bb12"],"text":"basis-map coefficients of A026729 \u2192 D"},"headline":"basis-map coefficients of A026729 \u2192 D","chain":null}},{"id":"l-e034a365791750df","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-e1393eff84da036c","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e1393eff84da036c","other":{"id":"n-e1393eff84da036c","name":"reversal of theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9"],["25"]],"card_rows":[["1"],["9"],["25"],["49"]],"row_sums_preview":["1","9","25","49","81","121","169","225"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-5ac7ff393747d6fc","e-191d9ac5d27e4874"],"text":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-e035f359f5a735da","kind":"fingerprint_related","node_a_id":"n-e30fce389f3d0ad4","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e30fce389f3d0ad4","other":{"id":"n-e30fce389f3d0ad4","name":"theta_plus_one_squared of theta_plus_one of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["0","-8"],["0","0","27"]],"card_rows":[["1"],["0","-8"],["0","0","27"],["0","0","0","-64"]],"row_sums_preview":["1","-8","27","-64","125","-216","343","-512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-d58e809d3242b4fa","e-3b6ad90c21eb255c"],"text":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-e05d1ab9470da33e","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-cf5644ef95030ccd","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-cf5644ef95030ccd","other":{"id":"n-cf5644ef95030ccd","name":"theta_plus_one of derivative of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","8"]],"card_rows":[["1"],["0","8"],["0","0","54"],["0","0","0","384"]],"row_sums_preview":["0","1","8","54","384","3000","25920","246960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["D","\u03b8+1"],"edges":["e-789a32df93c7cdb5","e-8a39cabd44d9ac0a"],"text":"A136572 \u2192 D \u2192 \u03b8+1"},"headline":"A136572 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-e06100a8bfd158da","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-55469d00c29c11d2","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-55469d00c29c11d2","other":{"id":"n-55469d00c29c11d2","name":"derivative of theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["0","18"],["0","0","0","100"]],"card_rows":[["0","18"],["0","0","0","100"],["0","0","0","0","0","294"],["0","0","0","0","0","0","0","648"]],"row_sums_preview":["0","18","100","294","648","1210","2028","3150"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["(\u03b8+1)\u00b2","D"],"edges":["e-5ac7ff393747d6fc","e-5c0e59154ea90f3d"],"text":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}},{"id":"l-e10927461641778f","kind":"fingerprint_related","node_a_id":"n-f9b18d63db7f99f5","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-f9b18d63db7f99f5","other":{"id":"n-f9b18d63db7f99f5","name":"derivative of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["0","4"]],"card_rows":[["-1"],["0","4"],["0","0","-18"],["0","0","0","96"]],"row_sums_preview":["0","-1","4","-18","96","-600","4320","-35280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["D"],"edges":["e-20618366745898a9"],"text":"basis-map coefficients of A021012 \u2192 D"},"headline":"basis-map coefficients of A021012 \u2192 D","chain":null}},{"id":"l-e16c840c8f0145bb","kind":"fingerprint_related","node_a_id":"n-e001d4321fcd493d","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e001d4321fcd493d","other":{"id":"n-e001d4321fcd493d","name":"gamma_polynomial of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11"],["121"]],"card_rows":[["1"],["11"],["121"],["1331"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["\u03b3"],"edges":["e-fb31a786d366dda0"],"text":"A038325 \u2192 \u03b3"},"headline":"A038325 \u2192 \u03b3","chain":null}},{"id":"l-e330dd07cd63cae4","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-5e2eb5cc04472685","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-5e2eb5cc04472685","other":{"id":"n-5e2eb5cc04472685","name":"derivative of theta_plus_one of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["4"]],"card_rows":[["4"],["0","18"],["0","0","48"],["0","0","0","100"]],"row_sums_preview":["0","0","4","18","48","100","180","294"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["\u03b8+1","D"],"edges":["e-0695c41826b28137","e-bd4fde74822026d7"],"text":"A132440 \u2192 \u03b8+1 \u2192 D"},"headline":"A132440 \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-e3468696b119773b","kind":"fingerprint_related","node_a_id":"n-791c020d1c792d56","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-791c020d1c792d56","other":{"id":"n-791c020d1c792d56","name":"derivative of theta_plus_one of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-2"],["0","6"]],"card_rows":[["-2"],["0","6"],["0","0","-12"],["0","0","0","20"]],"row_sums_preview":["0","-2","6","-12","20","-30","42","-56"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["\u03b8+1","D"],"edges":["e-5a10774a37bfecce","e-258c39471511ecf1"],"text":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 D"},"headline":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 D","chain":null}},{"id":"l-e8217ea618edc8ff","kind":"fingerprint_related","node_a_id":"n-e177181dad825c00","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-e177181dad825c00","other":{"id":"n-e177181dad825c00","name":"derivative of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","2"],["0","0","3"],["0","0","0","4"]],"card_rows":[["0","2"],["0","0","3"],["0","0","0","4"],["0","0","0","0","5"]],"row_sums_preview":["2","3","4","5","6","7","8","9"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["D"],"edges":["e-35fb5901b32bd10b"],"text":"basis-map coefficients of A112227 \u2192 D"},"headline":"basis-map coefficients of A112227 \u2192 D","chain":null}},{"id":"l-e861eff58c632b12","kind":"fingerprint_related","node_a_id":"n-eb46f673a485bcbf","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-eb46f673a485bcbf","other":{"id":"n-eb46f673a485bcbf","name":"theta_plus_one_squared of derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["0","8"]],"card_rows":[["-1"],["0","8"],["0","0","-27"],["0","0","0","64"]],"row_sums_preview":["0","-1","8","-27","64","-125","216","-343"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-c22d2d4fa11fd8cc","e-ec2df220d21387ed"],"text":"basis-map coefficients of A026729, shifted \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A026729, shifted \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-e94470c1880eae94","kind":"fingerprint_related","node_a_id":"n-72945891f8fb0616","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-72945891f8fb0616","other":{"id":"n-72945891f8fb0616","name":"theta_plus_one_squared of theta_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","27"],["0","0","0","64"],["0","0","0","0","125"]],"card_rows":[["0","0","27"],["0","0","0","64"],["0","0","0","0","125"],["0","0","0","0","0","216"]],"row_sums_preview":["27","64","125","216","343","512","729","1000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-c527a11d264443d6","e-54e9c1a3b6538492"],"text":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-e9f9bd0edff3e866","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-969fb1fba8487e2b","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-969fb1fba8487e2b","other":{"id":"n-969fb1fba8487e2b","name":"theta_plus_one_squared of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","16"],["0","0","81"]],"card_rows":[["1"],["0","16"],["0","0","81"],["0","0","0","256"]],"row_sums_preview":["1","16","81","256","625","1296","2401","4096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-3411eba2e020f0d3","e-a433a68f3f79f1fa"],"text":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-eadeed1a717357e9","kind":"fingerprint_related","node_a_id":"n-5a3397c669570bd5","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-5a3397c669570bd5","other":{"id":"n-5a3397c669570bd5","name":"reversal of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["4"]],"card_rows":[["1"],["2"],["4"],["8"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["reverse"],"edges":["e-3d2f34e0586880e9"],"text":"A130123 \u2192 reverse"},"headline":"A130123 \u2192 reverse","chain":null}},{"id":"l-ec05725a761acde5","kind":"fingerprint_related","node_a_id":"n-f3766a15d78768ea","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-f3766a15d78768ea","other":{"id":"n-f3766a15d78768ea","name":"theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","0","3"],["0","0","0","0","5"]],"card_rows":[["1"],["0","0","3"],["0","0","0","0","5"],["0","0","0","0","0","0","7"]],"row_sums_preview":["1","3","5","7","9","11","13","15"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["\u03b8+1"],"edges":["e-5ff024f1a04a7dc8"],"text":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1","chain":null}},{"id":"l-eccb6057ead9e917","kind":"fingerprint_related","node_a_id":"n-ccc86c3dcf2ac685","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-ccc86c3dcf2ac685","other":{"id":"n-ccc86c3dcf2ac685","name":"3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","4"],["0","0","7"]],"card_rows":[["1"],["0","4"],["0","0","7"],["0","0","0","10"]],"row_sums_preview":["1","4","7","10","13","16","19","22"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A131431"],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":[],"edges":[],"text":"A131431"},"headline":"A131431: 3n + 1 preceded by n zeros.","chain":null}},{"id":"l-ed232b89bca38541","kind":"fingerprint_related","node_a_id":"n-f6882297205c936c","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-f6882297205c936c","other":{"id":"n-f6882297205c936c","name":"theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","0","9"],["0","0","0","0","25"]],"card_rows":[["1"],["0","0","9"],["0","0","0","0","25"],["0","0","0","0","0","0","49"]],"row_sums_preview":["1","9","25","49","81","121","169","225"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["(\u03b8+1)\u00b2"],"edges":["e-5ac7ff393747d6fc"],"text":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-ed43eb9560236136","kind":"fingerprint_related","node_a_id":"n-eb8640ed213f9508","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-eb8640ed213f9508","other":{"id":"n-eb8640ed213f9508","name":"theta_plus_one of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","8"],["0","0","27"]],"card_rows":[["1"],["0","8"],["0","0","27"],["0","0","0","64"]],"row_sums_preview":["1","8","27","64","125","216","343","512"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["(\u03b8+1)\u00b2"],"edges":["e-048aefa516697baf"],"text":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-ee8bc28457deb9bb","kind":"fingerprint_related","node_a_id":"n-9c4445c28baba047","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-9c4445c28baba047","other":{"id":"n-9c4445c28baba047","name":"theta_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","3"],["0","0","0","4"],["0","0","0","0","5"]],"card_rows":[["0","0","3"],["0","0","0","4"],["0","0","0","0","5"],["0","0","0","0","0","6"]],"row_sums_preview":["3","4","5","6","7","8","9","10"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["\u03b8+1"],"edges":["e-972f6eabe8338070"],"text":"basis-map coefficients of A112227 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A112227 \u2192 \u03b8+1","chain":null}},{"id":"l-ef96dacbbf1bb346","kind":"fingerprint_related","node_a_id":"n-4b67dc586df1ece3","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-4b67dc586df1ece3","other":{"id":"n-4b67dc586df1ece3","name":"basis-map coefficients of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["0","0","1"]],"card_rows":[["0","0","1"],["0","0","0","3"],["0","0","0","0","6"],["0","0","0","0","0","10"]],"row_sums_preview":["0","0","1","3","6","10","15","21"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4b67dc586df1ece3","steps":[],"edges":[],"text":"basis-map coefficients of A237765"},"headline":"basis-map coefficients of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","chain":null}},{"id":"l-f0706dd7cd4835ea","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-c0ca6457f0f11aca","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-c0ca6457f0f11aca","other":{"id":"n-c0ca6457f0f11aca","name":"theta_plus_one of theta_plus_one of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","16"],["0","0","81"]],"card_rows":[["1"],["0","16"],["0","0","81"],["0","0","0","256"]],"row_sums_preview":["1","16","81","256","625","1296","2401","4096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["(\u03b8+1)\u00b2","\u03b8+1"],"edges":["e-048aefa516697baf","e-e5b06f8844ead11e"],"text":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1"},"headline":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 \u03b8+1","chain":null}},{"id":"l-f10f33ca165d1a54","kind":"fingerprint_related","node_a_id":"n-4d697803ecd7ecc7","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-4d697803ecd7ecc7","other":{"id":"n-4d697803ecd7ecc7","name":"reversal of theta_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["4"],["5"]],"card_rows":[["3"],["4"],["5"],["6"]],"row_sums_preview":["3","4","5","6","7","8","9","10"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["\u03b8+1","reverse"],"edges":["e-972f6eabe8338070","e-1005f159d7338c9b"],"text":"basis-map coefficients of A112227 \u2192 \u03b8+1 \u2192 reverse"},"headline":"basis-map coefficients of A112227 \u2192 \u03b8+1 \u2192 reverse","chain":null}},{"id":"l-f37effbda02b62df","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-a5ab08f961126261","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-a5ab08f961126261","other":{"id":"n-a5ab08f961126261","name":"reversal of theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4"],["9"]],"card_rows":[["1"],["-4"],["9"],["-16"]],"row_sums_preview":["1","-4","9","-16","25","-36","49","-64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-23b8fb76a1e0b114","e-aa4f1c527807dd39"],"text":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}},{"id":"l-f6caf68dd1b6b410","kind":"fingerprint_related","node_a_id":"n-4980330517acdfb6","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-4980330517acdfb6","other":{"id":"n-4980330517acdfb6","name":"gamma_polynomial of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15"],["45"]],"card_rows":[["3"],["15"],["45"],["105"]],"row_sums_preview":["3","15","45","105","210","378","630","990"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["\u03b3"],"edges":["e-4347e41634c98e26"],"text":"A344565 \u2192 \u03b3"},"headline":"A344565 \u2192 \u03b3","chain":null}},{"id":"l-f79cf0d103d285ab","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-818f66a9ea7208b3","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-818f66a9ea7208b3","other":{"id":"n-818f66a9ea7208b3","name":"theta_plus_one_squared of theta_plus_one_squared of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["0","32"]],"card_rows":[["1"],["0","32"],["0","0","243"],["0","0","0","1024"]],"row_sums_preview":["0","1","32","243","1024","3125","7776","16807"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["(\u03b8+1)\u00b2","(\u03b8+1)\u00b2"],"edges":["e-10cf23660c74d6bc","e-beb6333493147190"],"text":"A127648 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"A127648 \u2192 (\u03b8+1)\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-f9c9da515ef8a7ac","kind":"fingerprint_related","node_a_id":"n-300c5903ab08d510","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-300c5903ab08d510","other":{"id":"n-300c5903ab08d510","name":"derivative of Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["0","4"]],"card_rows":[["2"],["0","4"],["0","0","6"],["0","0","0","8"]],"row_sums_preview":["0","2","4","6","8","10","12","14"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-356713338f14325b","steps":["D"],"edges":["e-5f18485cf94c0ddc"],"text":"A136521 \u2192 D"},"headline":"A136521 \u2192 D","chain":null}},{"id":"l-faadf9212cf39d10","kind":"fingerprint_related","node_a_id":"n-be3d7a482cb99c2b","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-be3d7a482cb99c2b","other":{"id":"n-be3d7a482cb99c2b","name":"reversal of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4"],["7"]],"card_rows":[["1"],["4"],["7"],["10"]],"row_sums_preview":["1","4","7","10","13","16","19","22"],"regions":["(-inf, 0]","(-inf, -1]","[-1, 0]","[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["reverse"],"edges":["e-19e687b0ce9966ca"],"text":"A131431 \u2192 reverse"},"headline":"A131431 \u2192 reverse","chain":null}},{"id":"l-fc94016df10eceab","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-1a101d3d68ca8888","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-1a101d3d68ca8888","other":{"id":"n-1a101d3d68ca8888","name":"theta_plus_one_squared of derivative of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["0","32"]],"card_rows":[["2"],["0","32"],["0","0","216"],["0","0","0","1024"]],"row_sums_preview":["0","2","32","216","1024","4000","13824","43904"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-6151a42dffed124f","e-ab74e7e78d4ac881"],"text":"A130123 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A130123 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-fdbc430f29ba1c5b","kind":"fingerprint_related","node_a_id":"n-22781942b1af0ec6","node_b_id":"n-c6e3efe87a93824d","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-22781942b1af0ec6","other":{"id":"n-22781942b1af0ec6","name":"derivative of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["0","2"]],"card_rows":[["-1"],["0","2"],["0","0","-3"],["0","0","0","4"]],"row_sums_preview":["0","-1","2","-3","4","-5","6","-7"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["D"],"edges":["e-c22d2d4fa11fd8cc"],"text":"basis-map coefficients of A026729, shifted \u2192 D"},"headline":"basis-map coefficients of A026729, shifted \u2192 D","chain":null}},{"id":"l-fe3ab12c7ba8c938","kind":"fingerprint_related","node_a_id":"n-c6e3efe87a93824d","node_b_id":"n-36eafc78195eb9b6","fingerprint":"f6-edf6954a4cd8adfc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-edf6954a4cd8adfc"},"other_id":"n-36eafc78195eb9b6","other":{"id":"n-36eafc78195eb9b6","name":"theta_plus_one_squared of veronese_section_2_0 of A natural number operator.","first_index":0,"fingerprint":"f6-edf6954a4cd8adfc","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["0","20"]],"card_rows":[["1"],["2"],["0","20"],["0","24"]],"row_sums_preview":["1","2","20","24","81","90","208","224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f4c1dee76f5a68b2","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-bf26e3fe4e7c28ab","e-9ab066fb83f74a09"],"text":"A132774 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A132774 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 0;\n   2: 0;\n   3: 2;\n   4: 0, 6;\n   5: 0, 0, 12;\n   6: 0, 0,  0, 20;\n   7: 0, 0,  0,  0, 30;\n   8: 0, 0,  0,  0,  0, 42;\n   9: 0, 0,  0,  0,  0,  0, 56;\n  10: 0, 0,  0,  0,  0,  0,  0, 72;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (n-1)*x*P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1)*x*P[n - 1, x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((n - 1)*x*v[i - 1])/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-1bf99ad40f056d33","steps":["D","D"],"edges":["e-198d139196a8bb12","e-61cd49713ca3055b"],"text":"basis-map coefficients of A026729 \u2192 D \u2192 D"},"headline":"basis-map coefficients of A026729 \u2192 D \u2192 D","chain":null}