{"id":"n-f2187a6d363d8059","revision":0,"first_index":0,"fingerprint":"f6-2327141074f67bdf","name":"one_plus_derivative of theta_plus_one of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["8","8"],["0","54","27"],["0","0","192","64"],["0","0","0","500","125"],["0","0","0","0","1080","216"],["0","0","0","0","0","2058","343"],["0","0","0","0","0","0","3584","512"],["0","0","0","0","0","0","0","5832","729"],["0","0","0","0","0","0","0","0","9000","1000"],["0","0","0","0","0","0","0","0","0","13310","1331"],["0","0","0","0","0","0","0","0","0","0","19008","1728"],["0","0","0","0","0","0","0","0","0","0","0","26364","2197"],["0","0","0","0","0","0","0","0","0","0","0","0","35672","2744"],["0","0","0","0","0","0","0","0","0","0","0","0","0","47250","3375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","81","256","625","1296","2401","4096","6561","10000","14641","20736","28561","38416","50625"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["8","8"]],"recurrence":{"denominator":{"coeffs":[["0"],["0"],["0"],["-1"],["3"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","0","0"],["0","0","-1"],["0","0","-3"],["0","0","-3"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2"],["0","0"],["0","6"],["0","0"],["0","-6"],["0","0"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173209,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-34812ddb8e4a4262"]},{"id":173210,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-34812ddb8e4a4262"]},{"id":173211,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-34812ddb8e4a4262"]}],"region_exact":{},"incoming_edges":[{"id":"e-34812ddb8e4a4262","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-f2187a6d363d8059","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":"11049220c25fcb0125d95ec38b6911ee257db2ea577e410ba56fd5d268e3bcf1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eb8640ed213f9508","role":"sequence","name":"theta_plus_one of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-01de9d411fc04e18","kind":"fingerprint_related","node_a_id":"n-180e8c7ade26d4e8","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-180e8c7ade26d4e8","other":{"id":"n-180e8c7ade26d4e8","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","4"]],"card_rows":[["1"],["4","4"],["0","18","9"],["0","0","48","16"]],"row_sums_preview":["0","1","8","27","64","125","216","343"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["\u03b8+1","D+1"],"edges":["e-0695c41826b28137","e-d80cb4ba370acd0b"],"text":"A132440 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A132440 \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-0aaf11529cb426d6","kind":"fingerprint_related","node_a_id":"n-f7b0b5fcb17d3a59","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-f7b0b5fcb17d3a59","other":{"id":"n-f7b0b5fcb17d3a59","name":"one_plus_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.","first_index":1,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["2","2"]],"card_rows":[["1"],["2","2"],["0","6","3"],["0","0","12","4"]],"row_sums_preview":["0","1","4","9","16","25","36","49"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["D","D+1"],"edges":["e-198d139196a8bb12","e-ee27cab4e89b1aa0"],"text":"basis-map coefficients of A026729 \u2192 D \u2192 D+1"},"headline":"basis-map coefficients of A026729 \u2192 D \u2192 D+1","chain":null}},{"id":"l-0acbb01f11d39924","kind":"fingerprint_related","node_a_id":"n-b02759cc44b24b37","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-b02759cc44b24b37","other":{"id":"n-b02759cc44b24b37","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["0","2","1"]],"card_rows":[["1"],["1","1"],["0","2","1"],["0","0","3","1"]],"row_sums_preview":["1","2","3","4","5","6","7","8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["D+1"],"edges":["e-38c6c0f2ea6fe646"],"text":"basis-map coefficients of A026729 \u2192 D+1"},"headline":"basis-map coefficients of A026729 \u2192 D+1","chain":null}},{"id":"l-0b55a3e2e0691a71","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-099d6e3bb359cdfd","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-099d6e3bb359cdfd","other":{"id":"n-099d6e3bb359cdfd","name":"reversal of one_plus_derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["9","18"]],"card_rows":[["1"],["4","4"],["9","18"],["16","48"]],"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","D+1","reverse"],"edges":["e-06738ea5bb413735","e-2dc8b6d950a061ae","e-74dbe8dd90e4970e"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 reverse"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 reverse","chain":null}},{"id":"l-12d72293ac53939b","kind":"fingerprint_related","node_a_id":"n-e95078ee938f4693","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-e95078ee938f4693","other":{"id":"n-e95078ee938f4693","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6","6"],["0","24","12"],["0","0","60","20"]],"row_sums_preview":["0","0","2","12","36","80","150","252"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["D","D+1"],"edges":["e-1aec373ef809ab0b","e-da4b0fbd7b305369"],"text":"A127648 \u2192 D \u2192 D+1"},"headline":"A127648 \u2192 D \u2192 D+1","chain":null}},{"id":"l-23f376d380e5949b","kind":"fingerprint_related","node_a_id":"n-ee230c4e4a69ec67","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-ee230c4e4a69ec67","other":{"id":"n-ee230c4e4a69ec67","name":"reversal of one_plus_derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["7","14"]],"card_rows":[["1"],["4","4"],["7","14"],["10","30"]],"row_sums_preview":["1","8","21","40","65","96","133","176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D+1","reverse"],"edges":["e-133ff869421f85a2","e-0c25f089151136bb"],"text":"A131431 \u2192 D+1 \u2192 reverse"},"headline":"A131431 \u2192 D+1 \u2192 reverse","chain":null}},{"id":"l-24527e55d7a91c6d","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-c5b5d802fc81fcee","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-c5b5d802fc81fcee","other":{"id":"n-c5b5d802fc81fcee","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-2"],["0","12","6"]],"card_rows":[["1"],["-2","-2"],["0","12","6"],["0","0","-72","-24"]],"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","D+1"],"edges":["e-19f259e2ece869e6","e-2f7d583cedaf7376"],"text":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 D+1"},"headline":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-306e17979ca45b21","kind":"fingerprint_related","node_a_id":"n-91e8ef430c5dd0a0","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-91e8ef430c5dd0a0","other":{"id":"n-91e8ef430c5dd0a0","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-4"],["0","18","9"]],"card_rows":[["1"],["-4","-4"],["0","18","9"],["0","0","-48","-16"]],"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","D+1"],"edges":["e-23b8fb76a1e0b114","e-23efde9e23d44279"],"text":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D+1","chain":null}},{"id":"l-42eaf1434df6302d","kind":"fingerprint_related","node_a_id":"n-0ab341d7c1906bf7","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-0ab341d7c1906bf7","other":{"id":"n-0ab341d7c1906bf7","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["0","18","9"]],"card_rows":[["1"],["4","4"],["0","18","9"],["0","0","48","16"]],"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","D+1"],"edges":["e-3411eba2e020f0d3","e-f9b7eaee6f070951"],"text":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 D+1"},"headline":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 D+1","chain":null}},{"id":"l-44e53bc7976c1dad","kind":"fingerprint_related","node_a_id":"n-87f352aecee584cf","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-87f352aecee584cf","other":{"id":"n-87f352aecee584cf","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["2","2"]],"card_rows":[["-1"],["2","2"],["0","-6","-3"],["0","0","12","4"]],"row_sums_preview":["0","-1","4","-9","16","-25","36","-49"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["D","D+1"],"edges":["e-c22d2d4fa11fd8cc","e-e000a329e535a978"],"text":"basis-map coefficients of A026729, shifted \u2192 D \u2192 D+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 D \u2192 D+1","chain":null}},{"id":"l-56337362b4f06d09","kind":"fingerprint_related","node_a_id":"n-403f3ef85bba9ebc","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-403f3ef85bba9ebc","other":{"id":"n-403f3ef85bba9ebc","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-1"],["0","2","1"]],"card_rows":[["1"],["-1","-1"],["0","2","1"],["0","0","-3","-1"]],"row_sums_preview":["1","-2","3","-4","5","-6","7","-8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["D+1"],"edges":["e-dffa2a76d9a73ed0"],"text":"basis-map coefficients of A026729, shifted \u2192 D+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 D+1","chain":null}},{"id":"l-58967a486107f792","kind":"fingerprint_related","node_a_id":"n-7292b8be7150c934","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-7292b8be7150c934","other":{"id":"n-7292b8be7150c934","name":"one_plus_derivative of theta_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8"],["0","42","21"]],"card_rows":[["1"],["8","8"],["0","42","21"],["0","0","120","40"]],"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","D+1"],"edges":["e-5734274d793ba02f","e-480106f3757592b1"],"text":"A131431 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A131431 \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-696610f05fc57143","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-fac636095264228a","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-fac636095264228a","other":{"id":"n-fac636095264228a","name":"one_plus_derivative of theta_plus_one_squared of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","16"],["0","126","63"]],"card_rows":[["1"],["16","16"],["0","126","63"],["0","0","480","160"]],"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","D+1"],"edges":["e-792e4f12e133c3d5","e-82aa35904544443d"],"text":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 D+1"},"headline":"A131431 \u2192 (\u03b8+1)\u00b2 \u2192 D+1","chain":null}},{"id":"l-7188db5719a3e1ba","kind":"fingerprint_related","node_a_id":"n-d16445f4fa1fddba","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-d16445f4fa1fddba","other":{"id":"n-d16445f4fa1fddba","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["2","2"]],"card_rows":[["1"],["2","2"],["0","6","3"],["0","0","12","4"]],"row_sums_preview":["0","1","4","9","16","25","36","49"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["D+1"],"edges":["e-9e777fb0c714fbf5"],"text":"A132440 \u2192 D+1"},"headline":"A132440 \u2192 D+1","chain":null}},{"id":"l-76ff3f12bf276713","kind":"fingerprint_related","node_a_id":"n-2ec4ee15f07899eb","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-2ec4ee15f07899eb","other":{"id":"n-2ec4ee15f07899eb","name":"one_plus_derivative of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["0","12","6"]],"card_rows":[["1"],["3","3"],["0","12","6"],["0","0","30","10"]],"row_sums_preview":["1","6","18","40","75","126","196","288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["D+1"],"edges":["e-0110b668a29b77c5"],"text":"A127773 \u2192 D+1"},"headline":"A127773 \u2192 D+1","chain":null}},{"id":"l-7ed0e7d110561dba","kind":"fingerprint_related","node_a_id":"n-558dccd4023b101d","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-558dccd4023b101d","other":{"id":"n-558dccd4023b101d","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-2"],["0","6","3"]],"card_rows":[["1"],["-2","-2"],["0","6","3"],["0","0","-12","-4"]],"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","D+1"],"edges":["e-5a10774a37bfecce","e-e63c30eedc08205b"],"text":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 D+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-8039ede9729b1367","kind":"fingerprint_related","node_a_id":"n-2cd64ed988d7ab21","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-2cd64ed988d7ab21","other":{"id":"n-2cd64ed988d7ab21","name":"one_plus_derivative of derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["18","18"]],"card_rows":[["4"],["18","18"],["0","96","48"],["0","0","300","100"]],"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+1","D"],"edges":["e-06738ea5bb413735","e-2dc8b6d950a061ae","e-51e51676840655f6"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 D"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 D","chain":null}},{"id":"l-83d5549bd35fc467","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-1410ab806b8aaa2d","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-1410ab806b8aaa2d","other":{"id":"n-1410ab806b8aaa2d","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["0","24","12"]],"card_rows":[["1"],["4","4"],["0","24","12"],["0","0","96","32"]],"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","D+1"],"edges":["e-c327b1a68506ba1f","e-0c8e75cd1038245d"],"text":"A130123 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A130123 \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-8805ef390a365e35","kind":"fingerprint_related","node_a_id":"n-81d4142d81fa222d","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-81d4142d81fa222d","other":{"id":"n-81d4142d81fa222d","name":"one_plus_derivative of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["0","2","1"]],"card_rows":[["1"],["1","1"],["0","2","1"],["0","0","3","1"]],"row_sums_preview":["1","2","3","4","5","6","7","8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["D+1"],"edges":["e-63014276e02483f9"],"text":"A023531 \u2192 D+1"},"headline":"A023531 \u2192 D+1","chain":null}},{"id":"l-89a001e144ce00c3","kind":"fingerprint_related","node_a_id":"n-125b761c27788d9f","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-125b761c27788d9f","other":{"id":"n-125b761c27788d9f","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["2","2"]],"card_rows":[["-1"],["2","2"],["0","-6","-3"],["0","0","12","4"]],"row_sums_preview":["0","-1","4","-9","16","-25","36","-49"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["D","D+1"],"edges":["e-7b1079618314be79","e-eb6d7f98cca94cc8"],"text":"basis-map coefficients of A030528, shifted \u2192 D \u2192 D+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 D \u2192 D+1","chain":null}},{"id":"l-8be6104b55ff9ed5","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-d8e10938349e6e4d","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-d8e10938349e6e4d","other":{"id":"n-d8e10938349e6e4d","name":"one_plus_derivative of derivative of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","4"]],"card_rows":[["1"],["4","4"],["0","36","18"],["0","0","288","96"]],"row_sums_preview":["0","1","8","54","384","3000","25920","246960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["D","D+1"],"edges":["e-789a32df93c7cdb5","e-b44aacafa1e767c5"],"text":"A136572 \u2192 D \u2192 D+1"},"headline":"A136572 \u2192 D \u2192 D+1","chain":null}},{"id":"l-93cfddb24c6c504f","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-d7f59978973a7aed","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-d7f59978973a7aed","other":{"id":"n-d7f59978973a7aed","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8"],["0","72","36"]],"card_rows":[["1"],["8","8"],["0","72","36"],["0","0","384","128"]],"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","D+1"],"edges":["e-c89dffe6ace51b80","e-be4a1d7e73a22fb9"],"text":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 D+1"},"headline":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 D+1","chain":null}},{"id":"l-9c081af19b5a18dc","kind":"fingerprint_related","node_a_id":"n-e1f285b6913430e1","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-e1f285b6913430e1","other":{"id":"n-e1f285b6913430e1","name":"one_plus_derivative of derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["14","14"]],"card_rows":[["4"],["14","14"],["0","60","30"],["0","0","156","52"]],"row_sums_preview":["0","4","28","90","208","400","684","1078"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D+1","D"],"edges":["e-133ff869421f85a2","e-69820760f288c60b"],"text":"A131431 \u2192 D+1 \u2192 D"},"headline":"A131431 \u2192 D+1 \u2192 D","chain":null}},{"id":"l-9e557ada32d93adf","kind":"fingerprint_related","node_a_id":"n-c055169f8364a43f","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-c055169f8364a43f","other":{"id":"n-c055169f8364a43f","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","6"],["0","36","18"]],"card_rows":[["1"],["6","6"],["0","36","18"],["0","0","120","40"]],"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","D+1"],"edges":["e-9b1f1f612dc2ed52","e-3347a486013491cf"],"text":"A127773 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A127773 \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-9e6ed8e44874808e","kind":"fingerprint_related","node_a_id":"n-cbfe82ee301c996d","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-cbfe82ee301c996d","other":{"id":"n-cbfe82ee301c996d","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["0","8","4"]],"card_rows":[["1"],["2","2"],["0","8","4"],["0","0","24","8"]],"row_sums_preview":["1","4","12","32","80","192","448","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D+1"],"edges":["e-caef435f2360eea3"],"text":"A130123 \u2192 D+1"},"headline":"A130123 \u2192 D+1","chain":null}},{"id":"l-a4ae68612c6c89a7","kind":"fingerprint_related","node_a_id":"n-cda0343a0e78da93","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-cda0343a0e78da93","other":{"id":"n-cda0343a0e78da93","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","8"]],"card_rows":[["1"],["8","8"],["0","54","27"],["0","0","192","64"]],"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","D+1"],"edges":["e-10cf23660c74d6bc","e-43cf85ebc550359a"],"text":"A127648 \u2192 (\u03b8+1)\u00b2 \u2192 D+1"},"headline":"A127648 \u2192 (\u03b8+1)\u00b2 \u2192 D+1","chain":null}},{"id":"l-ac446e6f2312ba9e","kind":"fingerprint_related","node_a_id":"n-2abf8af6304c7215","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-2abf8af6304c7215","other":{"id":"n-2abf8af6304c7215","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["0","6","3"]],"card_rows":[["1"],["2","2"],["0","6","3"],["0","0","12","4"]],"row_sums_preview":["1","4","9","16","25","36","49","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["D+1"],"edges":["e-71be1602edcf5ea0"],"text":"image under \u03b8+1 \u2192 D+1"},"headline":"image under \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-b0b59b09332f8055","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-02c485e5596a5d6b","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-02c485e5596a5d6b","other":{"id":"n-02c485e5596a5d6b","name":"one_plus_derivative of theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["0","36","18"]],"card_rows":[["1"],["4","4"],["0","36","18"],["0","0","288","96"]],"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","D+1"],"edges":["e-76872140ee232c50","e-cd5d36f6d0fd951e"],"text":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 D+1"},"headline":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 D+1","chain":null}},{"id":"l-b32ad8dbed5ed086","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-10c14a72e8855913","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-10c14a72e8855913","other":{"id":"n-10c14a72e8855913","name":"one_plus_derivative of theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["0","12","6"]],"card_rows":[["1"],["2","2"],["0","12","6"],["0","0","72","24"]],"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","D+1"],"edges":["e-4dc8adfab569fa5c","e-e1ba84c8460dbd35"],"text":"A136572 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A136572 \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-c3d3b8ff0f63cd07","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-cb132c7465f0a373","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-cb132c7465f0a373","other":{"id":"n-cb132c7465f0a373","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","8"]],"card_rows":[["2"],["8","8"],["0","48","24"],["0","0","192","64"]],"row_sums_preview":["0","2","16","72","256","800","2304","6272"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D","D+1"],"edges":["e-6151a42dffed124f","e-5fc08bd250d0713a"],"text":"A130123 \u2192 D \u2192 D+1"},"headline":"A130123 \u2192 D \u2192 D+1","chain":null}},{"id":"l-cce54dc1061e31b9","kind":"fingerprint_related","node_a_id":"n-2a20d6494c8b5017","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-2a20d6494c8b5017","other":{"id":"n-2a20d6494c8b5017","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-2"],["0","6","3"]],"card_rows":[["1"],["-2","-2"],["0","6","3"],["0","0","-12","-4"]],"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","D+1"],"edges":["e-1d47beb63d4711b8","e-30037df1fc3089a3"],"text":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 D+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 D+1","chain":null}},{"id":"l-d47133bb94b8890f","kind":"fingerprint_related","node_a_id":"n-0f2ae50caf69594b","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-0f2ae50caf69594b","other":{"id":"n-0f2ae50caf69594b","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-4"],["0","18","9"]],"card_rows":[["1"],["-4","-4"],["0","18","9"],["0","0","-48","-16"]],"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","D+1"],"edges":["e-d58e809d3242b4fa","e-2492dac621d908e2"],"text":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 D+1","chain":null}},{"id":"l-d547c010284d81fb","kind":"fingerprint_related","node_a_id":"n-517084627d262afe","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-517084627d262afe","other":{"id":"n-517084627d262afe","name":"one_plus_derivative of Triangle read by rows, (n-1) zeros followed by 2^n - 1.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["0","14","7"]],"card_rows":[["1"],["3","3"],["0","14","7"],["0","0","45","15"]],"row_sums_preview":["1","6","21","60","155","378","889","2040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6394a556f7ad3cd4","steps":["D+1"],"edges":["e-65f9908356e1446e"],"text":"A131436 \u2192 D+1"},"headline":"A131436 \u2192 D+1","chain":null}},{"id":"l-ddb6378f64374742","kind":"fingerprint_related","node_a_id":"n-00cbe48522a066a8","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-00cbe48522a066a8","other":{"id":"n-00cbe48522a066a8","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-1"],["0","4","2"]],"card_rows":[["1"],["-1","-1"],["0","4","2"],["0","0","-18","-6"]],"row_sums_preview":["1","-2","6","-24","120","-720","5040","-40320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["D+1"],"edges":["e-224ed4e0bc1e1628"],"text":"basis-map coefficients of A021012 \u2192 D+1"},"headline":"basis-map coefficients of A021012 \u2192 D+1","chain":null}},{"id":"l-e48959e6061c51c8","kind":"fingerprint_related","node_a_id":"n-941fc2b5950fdf03","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-941fc2b5950fdf03","other":{"id":"n-941fc2b5950fdf03","name":"one_plus_derivative of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["0","4","2"]],"card_rows":[["1"],["1","1"],["0","4","2"],["0","0","18","6"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["D+1"],"edges":["e-2acff99f8d5ea860"],"text":"A136572 \u2192 D+1"},"headline":"A136572 \u2192 D+1","chain":null}},{"id":"l-ea166245ed73cf46","kind":"fingerprint_related","node_a_id":"n-4f63621ddd1bfac6","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-4f63621ddd1bfac6","other":{"id":"n-4f63621ddd1bfac6","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["12","12"]],"card_rows":[["3"],["12","12"],["0","60","30"],["0","0","180","60"]],"row_sums_preview":["0","3","24","90","240","525","1008","1764"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-511a2225fa2191c9","steps":["D","D+1"],"edges":["e-353b5c171e173f11","e-1969ebd269686ff1"],"text":"A127773 \u2192 D \u2192 D+1"},"headline":"A127773 \u2192 D \u2192 D+1","chain":null}},{"id":"l-f3e6436531dd558e","kind":"fingerprint_related","node_a_id":"n-df1048264f0e324d","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-df1048264f0e324d","other":{"id":"n-df1048264f0e324d","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-1"],["0","2","1"]],"card_rows":[["1"],["-1","-1"],["0","2","1"],["0","0","-3","-1"]],"row_sums_preview":["1","-2","3","-4","5","-6","7","-8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["D+1"],"edges":["e-806cb45b24538e1b"],"text":"basis-map coefficients of A030528, shifted \u2192 D+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 D+1","chain":null}},{"id":"l-f545694b2535b574","kind":"fingerprint_related","node_a_id":"n-bd4ec312ebef253e","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-bd4ec312ebef253e","other":{"id":"n-bd4ec312ebef253e","name":"one_plus_derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["0","14","7"]],"card_rows":[["1"],["4","4"],["0","14","7"],["0","0","30","10"]],"row_sums_preview":["1","8","21","40","65","96","133","176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D+1"],"edges":["e-133ff869421f85a2"],"text":"A131431 \u2192 D+1"},"headline":"A131431 \u2192 D+1","chain":null}},{"id":"l-f9297c8c85af684e","kind":"fingerprint_related","node_a_id":"n-f2187a6d363d8059","node_b_id":"n-831f650a4dd01253","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-831f650a4dd01253","other":{"id":"n-831f650a4dd01253","name":"one_plus_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":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["0","24","12"]],"card_rows":[["1"],["3","3"],["0","24","12"],["0","0","180","60"]],"row_sums_preview":["1","6","36","240","1800","15120","141120","1451520"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f7eaad3557182df","steps":["D+1"],"edges":["e-8ddcb030454781f8"],"text":"Hadamard-factor parent of A127773 \u2192 D+1"},"headline":"Hadamard-factor parent of A127773 \u2192 D+1","chain":null}},{"id":"l-fa71af556f82a1b3","kind":"fingerprint_related","node_a_id":"n-fc17ecb54217ecb0","node_b_id":"n-f2187a6d363d8059","fingerprint":"f6-2327141074f67bdf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2327141074f67bdf"},"other_id":"n-fc17ecb54217ecb0","other":{"id":"n-fc17ecb54217ecb0","name":"one_plus_derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-2327141074f67bdf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["0","18","9"]],"card_rows":[["1"],["4","4"],["0","18","9"],["0","0","48","16"]],"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","D+1"],"edges":["e-06738ea5bb413735","e-2dc8b6d950a061ae"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 8,  8;\n  2: 0, 54,  27;\n  3: 0,  0, 192,  64;\n  4: 0,  0,   0, 500,  125;\n  5: 0,  0,   0,   0, 1080,  216;\n  6: 0,  0,   0,   0,    0, 2058,  343;\n  7: 0,  0,   0,   0,    0,    0, 3584,  512;\n  8: 0,  0,   0,   0,    0,    0,    0, 5832,  729;\n  9: 0,  0,   0,   0,    0,    0,    0,    0, 9000, 1000;\n","recurrence_sentence":"The row polynomials satisfy (n^6-3*n^5+3*n^4-n^3)*P(n,x) = -(n^6+3*n^5+3*n^4+n^3)*x^2*P(n-2,x) + (2*n^6-6*n^4+6*n^2-2)*x*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 3*n^5 + 3*n^4 + n^3)*x^2*P[n - 2, x] + (2*n^6 - 6*n^4 + 6*n^2 - 2)*x*P[n - 1, x])/(n^6 - 3*n^5 + 3*n^4 - n^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 + 3*n^5 + 3*n^4 + n^3)*x^2*v[i - 2] + (2*n^6 - 6*n^4 + 6*n^2 - 2)*x*v[i - 1])/(n^6 - 3*n^5 + 3*n^4 - n^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7b3743413970d91c","steps":["(\u03b8+1)\u00b2","D+1"],"edges":["e-048aefa516697baf","e-34812ddb8e4a4262"],"text":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 D+1"},"headline":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 D+1","chain":null}