{"id":"n-3ca3059e0c8c21b8","revision":0,"first_index":0,"fingerprint":"f6-03d7b8417365c509","name":"derivative of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["4"],["128","32"],["3072","1536","192"],["65536","49152","12288","1024"],["1310720","1310720","491520","81920","5120"],["25165824","31457280","15728640","3932160","491520","24576"],["469762048","704643072","440401920","146800640","27525120","2752512","114688"],["8589934592","15032385536","11274289152","4697620480","1174405120","176160768","14680064","524288"],["154618822656","309237645312","270582939648","135291469824","42278584320","8455716864","1056964608","75497472","2359296"],["2748779069440","6184752906240","6184752906240","3607772528640","1352914698240","338228674560","56371445760","6039797760","377487360","10485760"],["48378511622144","120946279055360","136064563937280","90709709291520","39685497815040","11905649344512","2480343613440","354334801920","33218887680","1845493760","46137344"],["844424930131968","2322168557862912","2902710697328640","2177033022996480","1088516511498240","380980779024384","95245194756096","17008070492160","2126008811520","177167400960","8858370048","201326592"],["14636698788954112","43910096366862336","60376382504435712","50313652087029760","28301429298954240","11320571719581696","3301833418211328","707535732473856","110552458199040","12283606466560","921270484992","41875931136","872415232"],["252201579132747776","819655132181430272","1229482698272145408","1127025806749466624","704391129218416640","316976008148287488","105658669382762496","26414667345690624","4952750127316992","687881962127360","68788196212736","4690104287232","195421011968","3758096384"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","160","4800","128000","3200000","76800000","1792000000","40960000000","921600000000","20480000000000","450560000000000","9830400000000000","212992000000000000","4587520000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["16","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190930,"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-76ecb3d17f218949"]},{"id":190931,"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-76ecb3d17f218949"]},{"id":190932,"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-76ecb3d17f218949"]},{"id":190933,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-76ecb3d17f218949"]}],"region_exact":{},"incoming_edges":[{"id":"e-76ecb3d17f218949","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-3ca3059e0c8c21b8","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":"f9537ebb84b03270f3cba51462e7e54fc7b8dab1251cb59c17cbdbba605d22a8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-21b46279e9abe0a4","role":"sequence","name":"basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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-07bb42d8f77aaafb","kind":"fingerprint_related","node_a_id":"n-15850e47aea7175f","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-15850e47aea7175f","other":{"id":"n-15850e47aea7175f","name":"derivative of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["24","6"],["192","96","12"],["1280","960","240","20"]],"row_sums_preview":["0","0","2","30","300","2500","18750","131250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D","D"],"edges":["e-62aaaa59f91d42c1","e-61fe88a6e816cd1d"],"text":"A038231 \u2192 D \u2192 D"},"headline":"A038231 \u2192 D \u2192 D","chain":null}},{"id":"l-0a447435b07b89a5","kind":"fingerprint_related","node_a_id":"n-146f1cd501c36740","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-146f1cd501c36740","other":{"id":"n-146f1cd501c36740","name":"basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","8","16"],["1","12","48","64"],["1","16","96","256","256"]],"card_rows":[["1","8","16"],["1","12","48","64"],["1","16","96","256","256"],["1","20","160","640","1280","1024"]],"row_sums_preview":["25","125","625","3125","15625","78125","390625","1953125"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611"],"edges":["e-b91bf4c52fd06745"],"text":"basis-map coefficients of A112227 \u2192 basis A013611"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611","chain":null}},{"id":"l-0f4a30230d57044a","kind":"fingerprint_related","node_a_id":"n-3dcaebc50e901d48","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-3dcaebc50e901d48","other":{"id":"n-3dcaebc50e901d48","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","1"],["16","8","1"]],"card_rows":[["1"],["4","1"],["16","8","1"],["64","48","12","1"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038231"],"derivation":{"root":"n-3dcaebc50e901d48","steps":[],"edges":[],"text":"A038231"},"headline":"A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","chain":null}},{"id":"l-1158660c5af0a663","kind":"fingerprint_related","node_a_id":"n-1aa5741cc40f99e1","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-1aa5741cc40f99e1","other":{"id":"n-1aa5741cc40f99e1","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["32","8"]],"card_rows":[["2"],["32","8"],["384","192","24"],["4096","3072","768","64"]],"row_sums_preview":["0","2","40","600","8000","100000","1200000","14000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013609","D"],"edges":["e-13f16c9d95bb60f6","e-163a33265ccce32f"],"text":"A027466 \u2192 basis A013609 \u2192 D"},"headline":"A027466 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-12153df25376876b","kind":"fingerprint_related","node_a_id":"n-872d1ad8224c1da6","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-872d1ad8224c1da6","other":{"id":"n-872d1ad8224c1da6","name":"basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["2","16","32"]],"card_rows":[["1"],["1","4"],["2","16","32"],["6","72","288","384"]],"row_sums_preview":["1","5","50","750","15000","375000","11250000","393750000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611"],"edges":["e-3656191e13afb7f3"],"text":"A136572 \u2192 basis A013611"},"headline":"A136572 \u2192 basis A013611","chain":null}},{"id":"l-18307dd9875ff59f","kind":"fingerprint_related","node_a_id":"n-6e0864853f49fb26","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-6e0864853f49fb26","other":{"id":"n-6e0864853f49fb26","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-4"],["2","16","32"]],"card_rows":[["1"],["-1","-4"],["2","16","32"],["-6","-72","-288","-384"]],"row_sums_preview":["1","-5","50","-750","15000","-375000","11250000","-393750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A013611"],"edges":["e-8a1316f2e0d28647"],"text":"basis-map coefficients of A021012 \u2192 basis A013611"},"headline":"basis-map coefficients of A021012 \u2192 basis A013611","chain":null}},{"id":"l-1c525acb63ece5fc","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-8dae71118319cf76","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-8dae71118319cf76","other":{"id":"n-8dae71118319cf76","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["2","8"]],"card_rows":[["1"],["2","8"],["3","24","48"],["4","48","192","256"]],"row_sums_preview":["0","1","10","75","500","3125","18750","109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["D","basis A013611"],"edges":["e-198d139196a8bb12","e-8bb9cbbc61946631"],"text":"basis-map coefficients of A026729 \u2192 D \u2192 basis A013611"},"headline":"basis-map coefficients of A026729 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-1fce057edd6551ff","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-e52386b4bc8a55e4","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-e52386b4bc8a55e4","other":{"id":"n-e52386b4bc8a55e4","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["128","32"],["3072","1536","192"],["65536","49152","12288","1024"]],"card_rows":[["128","32"],["3072","1536","192"],["65536","49152","12288","1024"],["1310720","1310720","491520","81920","5120"]],"row_sums_preview":["160","4800","128000","3200000","76800000","1792000000","40960000000","921600000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","D"],"edges":["e-7b68ca495e98f7ea","e-2bdc068f6320ee7b"],"text":"A038330 \u2192 basis A007318 \u2192 D"},"headline":"A038330 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-28eece3ccbf1c617","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-30d2ffb507fad587","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-30d2ffb507fad587","other":{"id":"n-30d2ffb507fad587","name":"basis_map_A013611 of theta_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","32"],["21","168","336"]],"card_rows":[["1"],["8","32"],["21","168","336"],["40","480","1920","2560"]],"row_sums_preview":["1","40","525","5000","40625","300000","2078125","13750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["\u03b8+1","basis A013611"],"edges":["e-5734274d793ba02f","e-f287e20a316cb76d"],"text":"A131431 \u2192 \u03b8+1 \u2192 basis A013611"},"headline":"A131431 \u2192 \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-29040307bdf081ca","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-9de0875f066ebfc9","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-9de0875f066ebfc9","other":{"id":"n-9de0875f066ebfc9","name":"basis_map_A013611 of Right shift operator generating 1's in shifted spaces.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","8","16"]],"row_sums_preview":["1","1","5","25","125","625","3125","15625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d2ea2ff0a779ba6e","steps":["basis A013611"],"edges":["e-55ed28e2cac63f49"],"text":"A129186 \u2192 basis A013611"},"headline":"A129186 \u2192 basis A013611","chain":null}},{"id":"l-2e79220201dc6baa","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-1d109880d7daa4bc","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-1d109880d7daa4bc","other":{"id":"n-1d109880d7daa4bc","name":"reversal of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16"],["16","128","256"]],"card_rows":[["1"],["4","16"],["16","128","256"],["64","768","3072","4096"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","reverse"],"edges":["e-9c08db68889738c4","e-5da1211147e9d2da"],"text":"A027467 \u2192 basis A013611 \u2192 reverse"},"headline":"A027467 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-3130a2b9e1fff177","kind":"fingerprint_related","node_a_id":"n-1e5a38872f244688","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-1e5a38872f244688","other":{"id":"n-1e5a38872f244688","name":"derivative of derivative of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["32"]],"card_rows":[["32"],["96","384"],["192","1536","3072"],["320","3840","15360","20480"]],"row_sums_preview":["0","0","32","480","4800","40000","300000","2100000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["D","D"],"edges":["e-b04b02754d5cebbf","e-273e1dad22561e44"],"text":"A013611 \u2192 D \u2192 D"},"headline":"A013611 \u2192 D \u2192 D","chain":null}},{"id":"l-3496c3ab55c30041","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-5d5c2b8d94adf719","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-5d5c2b8d94adf719","other":{"id":"n-5d5c2b8d94adf719","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","32"],["36","288","576"]],"card_rows":[["1"],["8","32"],["36","288","576"],["128","1536","6144","8192"]],"row_sums_preview":["1","40","900","16000","250000","3600000","49000000","640000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["(\u03b8+1)\u00b2","basis A013611"],"edges":["e-c89dffe6ace51b80","e-d5b69f835b199262"],"text":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611"},"headline":"A130123 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611","chain":null}},{"id":"l-38f246ce6fd92b34","kind":"fingerprint_related","node_a_id":"n-615df6ea54d3bbef","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-615df6ea54d3bbef","other":{"id":"n-615df6ea54d3bbef","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","72","9"],["1728","1296","324","27"],["20736","20736","7776","1296","81"]],"card_rows":[["144","72","9"],["1728","1296","324","27"],["20736","20736","7776","1296","81"],["248832","311040","155520","38880","4860","243"]],"row_sums_preview":["225","3375","50625","759375","11390625","170859375","2562890625","38443359375"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":[],"edges":[],"text":"A038329"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","chain":null}},{"id":"l-3b2f2b5308a390f0","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-4804690590fbfb7a","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-4804690590fbfb7a","other":{"id":"n-4804690590fbfb7a","name":"basis_map_A013611 of Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["2","16","32"]],"card_rows":[["1"],["2","8"],["2","16","32"],["2","24","96","128"]],"row_sums_preview":["1","10","50","250","1250","6250","31250","156250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-356713338f14325b","steps":["basis A013611"],"edges":["e-929d7edb4c8e12ea"],"text":"A136521 \u2192 basis A013611"},"headline":"A136521 \u2192 basis A013611","chain":null}},{"id":"l-3c0fef475790cfbd","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-0ff5956186d03d47","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-0ff5956186d03d47","other":{"id":"n-0ff5956186d03d47","name":"basis_map_A013611 of Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["2","16","32"]],"card_rows":[["1"],["1","4"],["2","16","32"],["4","48","192","256"]],"row_sums_preview":["1","5","50","500","5000","50000","500000","5000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a895a528c93c18ea","steps":["basis A013611"],"edges":["e-422c4376a5ac0423"],"text":"A134309 \u2192 basis A013611"},"headline":"A134309 \u2192 basis A013611","chain":null}},{"id":"l-3d24266f5eacd9a3","kind":"fingerprint_related","node_a_id":"n-4826ffe2fa9d46c8","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-4826ffe2fa9d46c8","other":{"id":"n-4826ffe2fa9d46c8","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16"],["16","128","256"]],"card_rows":[["1"],["4","16"],["16","128","256"],["64","768","3072","4096"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-82b80af703a32f5e","e-27c8579fa1593a9f"],"text":"A013610 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A013610 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-414dea9f24ea314a","kind":"fingerprint_related","node_a_id":"n-8547d44959e7b9a0","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-8547d44959e7b9a0","other":{"id":"n-8547d44959e7b9a0","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-4"],["1","8","16"]],"card_rows":[["1"],["-1","-4"],["1","8","16"],["-1","-12","-48","-64"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013611"],"edges":["e-95b12e1c49fd2aa5"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013611"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013611","chain":null}},{"id":"l-42821695ebbc2f6a","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-21ab6fc21a4c4348","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-21ab6fc21a4c4348","other":{"id":"n-21ab6fc21a4c4348","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["72","18"],["1296","648","81"],["20736","15552","3888","324"]],"card_rows":[["72","18"],["1296","648","81"],["20736","15552","3888","324"],["311040","311040","116640","19440","1215"]],"row_sums_preview":["90","2025","40500","759375","13668750","239203125","4100625000","69198046875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["D"],"edges":["e-56784c43cb830e8e"],"text":"A038329 \u2192 D"},"headline":"A038329 \u2192 D","chain":null}},{"id":"l-4295e60896e5e6ba","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-0990a78160b6cbf7","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-0990a78160b6cbf7","other":{"id":"n-0990a78160b6cbf7","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6","24"],["12","96","192"],["20","240","960","1280"]],"row_sums_preview":["0","0","2","30","300","2500","18750","131250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["D","basis A013611"],"edges":["e-1aec373ef809ab0b","e-680362cc8865865c"],"text":"A127648 \u2192 D \u2192 basis A013611"},"headline":"A127648 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-430047e83084dfdc","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-ce5b0c4643351ea7","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-ce5b0c4643351ea7","other":{"id":"n-ce5b0c4643351ea7","name":"basis_map_A013611 of derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["18","72"]],"card_rows":[["4"],["18","72"],["48","384","768"],["100","1200","4800","6400"]],"row_sums_preview":["0","4","90","1200","12500","112500","918750","7000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D","basis A013611"],"edges":["e-06738ea5bb413735","e-864c354a3903aafb","e-f615883c5fdfd7df"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D \u2192 basis A013611"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-498619abe1093955","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-7514099f2f191994","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-7514099f2f191994","other":{"id":"n-7514099f2f191994","name":"basis_map_A013611 of Triangle read by rows, (n-1) zeros followed by 2^n - 1.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","12"],["7","56","112"]],"card_rows":[["1"],["3","12"],["7","56","112"],["15","180","720","960"]],"row_sums_preview":["1","15","175","1875","19375","196875","1984375","19921875"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6394a556f7ad3cd4","steps":["basis A013611"],"edges":["e-6345331e30ed4cd3"],"text":"A131436 \u2192 basis A013611"},"headline":"A131436 \u2192 basis A013611","chain":null}},{"id":"l-49d2965a7e41237c","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-4212eedcfcdfdfcc","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-4212eedcfcdfdfcc","other":{"id":"n-4212eedcfcdfdfcc","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16"],["12","96","192"]],"card_rows":[["1"],["4","16"],["12","96","192"],["32","384","1536","2048"]],"row_sums_preview":["1","20","300","4000","50000","600000","7000000","80000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["\u03b8+1","basis A013611"],"edges":["e-c327b1a68506ba1f","e-a1f143f6f237ba63"],"text":"A130123 \u2192 \u03b8+1 \u2192 basis A013611"},"headline":"A130123 \u2192 \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-4de5b6265aab702a","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-e834f79d5cb062d1","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-e834f79d5cb062d1","other":{"id":"n-e834f79d5cb062d1","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","72","144"],["27","324","1296","1728"],["81","1296","7776","20736","20736"]],"card_rows":[["9","72","144"],["27","324","1296","1728"],["81","1296","7776","20736","20736"],["243","4860","38880","155520","311040","248832"]],"row_sums_preview":["225","3375","50625","759375","11390625","170859375","2562890625","38443359375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["reverse"],"edges":["e-be644704bcbcfe05"],"text":"A038329 \u2192 reverse"},"headline":"A038329 \u2192 reverse","chain":null}},{"id":"l-4fc852a909a987c3","kind":"fingerprint_related","node_a_id":"n-ee879560f84ab2b6","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-ee879560f84ab2b6","other":{"id":"n-ee879560f84ab2b6","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","128","16"],["4096","3072","768","64"],["65536","65536","24576","4096","256"]],"card_rows":[["256","128","16"],["4096","3072","768","64"],["65536","65536","24576","4096","256"],["1048576","1310720","655360","163840","20480","1024"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318"],"edges":["e-7b68ca495e98f7ea"],"text":"A038330 \u2192 basis A007318"},"headline":"A038330 \u2192 basis A007318","chain":null}},{"id":"l-5233c4ef785bdb52","kind":"fingerprint_related","node_a_id":"n-bcadabd274f97937","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-bcadabd274f97937","other":{"id":"n-bcadabd274f97937","name":"derivative of basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8"],["32","128"]],"card_rows":[["8"],["32","128"],["96","768","1536"],["256","3072","12288","16384"]],"row_sums_preview":["0","8","160","2400","32000","400000","4800000","56000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","D"],"edges":["e-d7d2422771947afc","e-72cef0f2a400ad6d"],"text":"A130123 \u2192 basis A013611 \u2192 D"},"headline":"A130123 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-551463c23bcb9451","kind":"fingerprint_related","node_a_id":"n-2e0409c0014ec240","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-2e0409c0014ec240","other":{"id":"n-2e0409c0014ec240","name":"Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["1","8","16"]],"card_rows":[["1"],["1","4"],["1","8","16"],["1","12","48","64"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013611"],"derivation":{"root":"n-2e0409c0014ec240","steps":[],"edges":[],"text":"A013611"},"headline":"A013611: Triangle of coefficients in expansion of (1+4x)^n.","chain":null}},{"id":"l-58eb9ae6ee023c19","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-162098c126ccbc3b","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-162098c126ccbc3b","other":{"id":"n-162098c126ccbc3b","name":"basis_map_A013611 of theta_plus_one_squared of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","72","144"],["16","192","768","1024"],["25","400","2400","6400","6400"]],"card_rows":[["9","72","144"],["16","192","768","1024"],["25","400","2400","6400","6400"],["36","720","5760","23040","46080","36864"]],"row_sums_preview":["225","2000","15625","112500","765625","5000000","31640625","195312500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["(\u03b8+1)\u00b2","basis A013611"],"edges":["e-c527a11d264443d6","e-fdaf60cb8dbf5427"],"text":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611"},"headline":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611","chain":null}},{"id":"l-59076fc7c44df6da","kind":"fingerprint_related","node_a_id":"n-a8775879b5c04126","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-a8775879b5c04126","other":{"id":"n-a8775879b5c04126","name":"shift_x_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["12","3"],["48","24","3"]],"card_rows":[["3"],["12","3"],["48","24","3"],["192","144","36","3"]],"row_sums_preview":["3","15","75","375","1875","9375","46875","234375"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A007318","basis A007318"],"edges":["e-18f137fab6db112d","e-0d0c855614bcbe4c"],"text":"A120910 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A120910 \u2192 basis A007318 \u2192 basis A007318","chain":null}},{"id":"l-6202352a5a4097ed","kind":"fingerprint_related","node_a_id":"n-fdcf3b7c800e805f","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-fdcf3b7c800e805f","other":{"id":"n-fdcf3b7c800e805f","name":"derivative of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["8","32"],["12","96","192"],["16","192","768","1024"]],"card_rows":[["8","32"],["12","96","192"],["16","192","768","1024"],["20","320","1920","5120","5120"]],"row_sums_preview":["40","300","2000","12500","75000","437500","2500000","14062500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611","D"],"edges":["e-b91bf4c52fd06745","e-10492c35b036258c"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 D"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-62af99c9bb21adde","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-43d60147a8c06e7d","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-43d60147a8c06e7d","other":{"id":"n-43d60147a8c06e7d","name":"basis_map_A013611 of theta_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3","24","48"],["4","48","192","256"],["5","80","480","1280","1280"]],"card_rows":[["3","24","48"],["4","48","192","256"],["5","80","480","1280","1280"],["6","120","960","3840","7680","6144"]],"row_sums_preview":["75","500","3125","18750","109375","625000","3515625","19531250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["\u03b8+1","basis A013611"],"edges":["e-972f6eabe8338070","e-1912707a73bc94cf"],"text":"basis-map coefficients of A112227 \u2192 \u03b8+1 \u2192 basis A013611"},"headline":"basis-map coefficients of A112227 \u2192 \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-62b5f0edcf9917e9","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-9b654d570e0f3f5e","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-9b654d570e0f3f5e","other":{"id":"n-9b654d570e0f3f5e","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-8"],["3","24","48"]],"card_rows":[["1"],["-2","-8"],["3","24","48"],["-4","-48","-192","-256"]],"row_sums_preview":["1","-10","75","-500","3125","-18750","109375","-625000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["\u03b8+1","basis A013611"],"edges":["e-1d47beb63d4711b8","e-429f860293c1f430"],"text":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 basis A013611"},"headline":"basis-map coefficients of A026729, shifted \u2192 \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-63e220756c9fe61c","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-8a8e1f3c56333ad5","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-8a8e1f3c56333ad5","other":{"id":"n-8a8e1f3c56333ad5","name":"basis_map_A013611 of theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["6","48","96"]],"card_rows":[["1"],["2","8"],["6","48","96"],["24","288","1152","1536"]],"row_sums_preview":["1","10","150","3000","75000","2250000","78750000","3150000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["\u03b8+1","basis A013611"],"edges":["e-4dc8adfab569fa5c","e-a6797193e7d36cb2"],"text":"A136572 \u2192 \u03b8+1 \u2192 basis A013611"},"headline":"A136572 \u2192 \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-684169688dc66d93","kind":"fingerprint_related","node_a_id":"n-be5fbaa0cf892ed1","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-be5fbaa0cf892ed1","other":{"id":"n-be5fbaa0cf892ed1","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","3"],["144","72","9"]],"card_rows":[["1"],["12","3"],["144","72","9"],["1728","1296","324","27"]],"row_sums_preview":["1","15","225","3375","50625","759375","11390625","170859375"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A007318","basis A013610"],"edges":["e-34d290d6207a0155","e-48f172fccaafebe8"],"text":"A038303 \u2192 basis A007318 \u2192 basis A013610"},"headline":"A038303 \u2192 basis A007318 \u2192 basis A013610","chain":null}},{"id":"l-692fd90c303f842f","kind":"fingerprint_related","node_a_id":"n-7a0aecafcb4a073f","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-7a0aecafcb4a073f","other":{"id":"n-7a0aecafcb4a073f","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-4"],["1","8","16"]],"card_rows":[["1"],["-1","-4"],["1","8","16"],["-1","-12","-48","-64"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013611"],"edges":["e-f6ed0d45ba75837f"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013611"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013611","chain":null}},{"id":"l-74d5dec6fec0ae5f","kind":"fingerprint_related","node_a_id":"n-84190c5a93dd74bf","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-84190c5a93dd74bf","other":{"id":"n-84190c5a93dd74bf","name":"reversal of basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","1"],["16","8","1"]],"card_rows":[["1"],["4","1"],["16","8","1"],["64","48","12","1"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611","reverse"],"edges":["e-f93ad378f9476e41","e-3c895899425b2755"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 reverse"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-760aede5e435e39c","kind":"fingerprint_related","node_a_id":"n-6f3f150f98e11d9a","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-6f3f150f98e11d9a","other":{"id":"n-6f3f150f98e11d9a","name":"reversal of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["16","8","1"],["64","48","12","1"],["256","256","96","16","1"]],"card_rows":[["16","8","1"],["64","48","12","1"],["256","256","96","16","1"],["1024","1280","640","160","20","1"]],"row_sums_preview":["25","125","625","3125","15625","78125","390625","1953125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611","reverse"],"edges":["e-b91bf4c52fd06745","e-2f2f964e00b8d7b1"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 reverse"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-7b0dd7a76af3d96f","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-2779a75fc60be9a6","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-2779a75fc60be9a6","other":{"id":"n-2779a75fc60be9a6","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-16"],["9","72","144"]],"card_rows":[["1"],["-4","-16"],["9","72","144"],["-16","-192","-768","-1024"]],"row_sums_preview":["1","-20","225","-2000","15625","-112500","765625","-5000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["(\u03b8+1)\u00b2","basis A013611"],"edges":["e-23b8fb76a1e0b114","e-bf1bcfa6d864a6cb"],"text":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611"},"headline":"basis-map coefficients of A026729, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611","chain":null}},{"id":"l-8499364c33e56ff9","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-9a0e13bbdbbf886c","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-9a0e13bbdbbf886c","other":{"id":"n-9a0e13bbdbbf886c","name":"basis_map_A013611 of theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["3","24","48"]],"card_rows":[["1"],["2","8"],["3","24","48"],["4","48","192","256"]],"row_sums_preview":["1","10","75","500","3125","18750","109375","625000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["basis A013611"],"edges":["e-dd7b96e2d01eda0d"],"text":"image under \u03b8+1 \u2192 basis A013611"},"headline":"image under \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-88624e07df02e2a2","kind":"fingerprint_related","node_a_id":"n-be86902a3ebfde1b","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-be86902a3ebfde1b","other":{"id":"n-be86902a3ebfde1b","name":"derivative of basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["8","32"]],"card_rows":[["4"],["8","32"],["12","96","192"],["16","192","768","1024"]],"row_sums_preview":["0","4","40","300","2000","12500","75000","437500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611","D"],"edges":["e-f93ad378f9476e41","e-842c2ef513433d75"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 D"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-8ed2461f5a476230","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-d74992e60bc95ea4","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-d74992e60bc95ea4","other":{"id":"n-d74992e60bc95ea4","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-8"],["3","24","48"]],"card_rows":[["1"],["-2","-8"],["3","24","48"],["-4","-48","-192","-256"]],"row_sums_preview":["1","-10","75","-500","3125","-18750","109375","-625000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["\u03b8+1","basis A013611"],"edges":["e-5a10774a37bfecce","e-72fcfaa57b9b3887"],"text":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 basis A013611"},"headline":"basis-map coefficients of A030528, shifted \u2192 \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-8f3f7c886034642c","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-c461466144e48481","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-c461466144e48481","other":{"id":"n-c461466144e48481","name":"basis_map_A013611 of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16"],["7","56","112"]],"card_rows":[["1"],["4","16"],["7","56","112"],["10","120","480","640"]],"row_sums_preview":["1","20","175","1250","8125","50000","296875","1718750"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["basis A013611"],"edges":["e-26078f28354a5abc"],"text":"A131431 \u2192 basis A013611"},"headline":"A131431 \u2192 basis A013611","chain":null}},{"id":"l-92a148ac0d817f1f","kind":"fingerprint_related","node_a_id":"n-88c0cd8da579d1de","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-88c0cd8da579d1de","other":{"id":"n-88c0cd8da579d1de","name":"reversal of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","1"],["32","16","2"]],"card_rows":[["1"],["4","1"],["32","16","2"],["384","288","72","6"]],"row_sums_preview":["1","5","50","750","15000","375000","11250000","393750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","reverse"],"edges":["e-3656191e13afb7f3","e-4179fa8b8aab55ae"],"text":"A136572 \u2192 basis A013611 \u2192 reverse"},"headline":"A136572 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-95f70ac8e5fb977c","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-0602630496894628","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-0602630496894628","other":{"id":"n-0602630496894628","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["2","8"]],"card_rows":[["1"],["2","8"],["3","24","48"],["4","48","192","256"]],"row_sums_preview":["0","1","10","75","500","3125","18750","109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["basis A013611"],"edges":["e-23f371b7f38c0ebd"],"text":"A132440 \u2192 basis A013611"},"headline":"A132440 \u2192 basis A013611","chain":null}},{"id":"l-9b0d5ff3e773a74a","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-c08d784e2bce5ca1","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-c08d784e2bce5ca1","other":{"id":"n-c08d784e2bce5ca1","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","32"]],"card_rows":[["2"],["8","32"],["24","192","384"],["64","768","3072","4096"]],"row_sums_preview":["0","2","40","600","8000","100000","1200000","14000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D","basis A013611"],"edges":["e-6151a42dffed124f","e-d3e425406fb0eb0a"],"text":"A130123 \u2192 D \u2192 basis A013611"},"headline":"A130123 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-a63a6e7bd68f7f24","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-24b741eba413aeed","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-24b741eba413aeed","other":{"id":"n-24b741eba413aeed","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-8"],["6","48","96"]],"card_rows":[["1"],["-2","-8"],["6","48","96"],["-24","-288","-1152","-1536"]],"row_sums_preview":["1","-10","150","-3000","75000","-2250000","78750000","-3150000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["\u03b8+1","basis A013611"],"edges":["e-19f259e2ece869e6","e-d866df21d4355ff2"],"text":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 basis A013611"},"headline":"basis-map coefficients of A021012 \u2192 \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-acfbd47a45af2c24","kind":"fingerprint_related","node_a_id":"n-1e5aa25413f55f87","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-1e5aa25413f55f87","other":{"id":"n-1e5aa25413f55f87","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["1","8","16"]],"card_rows":[["1"],["1","4"],["1","8","16"],["1","12","48","64"]],"row_sums_preview":["1","5","25","125","625","3125","15625","78125"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611"],"edges":["e-f93ad378f9476e41"],"text":"basis-map coefficients of A026729 \u2192 basis A013611"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611","chain":null}},{"id":"l-aebf909f45ac466f","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-04caad966d625ca5","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-04caad966d625ca5","other":{"id":"n-04caad966d625ca5","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","16"]],"card_rows":[["1"],["4","16"],["9","72","144"],["16","192","768","1024"]],"row_sums_preview":["0","1","20","225","2000","15625","112500","765625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["\u03b8+1","basis A013611"],"edges":["e-0695c41826b28137","e-815114bf7cb9cebf"],"text":"A132440 \u2192 \u03b8+1 \u2192 basis A013611"},"headline":"A132440 \u2192 \u03b8+1 \u2192 basis A013611","chain":null}},{"id":"l-b235695c71abf305","kind":"fingerprint_related","node_a_id":"n-9b2827e5f6c3d47c","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-9b2827e5f6c3d47c","other":{"id":"n-9b2827e5f6c3d47c","name":"derivative of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["16","64"]],"card_rows":[["4"],["16","64"],["72","576","1152"],["384","4608","18432","24576"]],"row_sums_preview":["0","4","80","1800","48000","1500000","54000000","2205000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","D"],"edges":["e-3656191e13afb7f3","e-e1579b67b7d69528"],"text":"A136572 \u2192 basis A013611 \u2192 D"},"headline":"A136572 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-ba5d52c52074e625","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-d7735703c5b80c21","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-d7735703c5b80c21","other":{"id":"n-d7735703c5b80c21","name":"basis_map_A013611 of derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["14","56"]],"card_rows":[["4"],["14","56"],["30","240","480"],["52","624","2496","3328"]],"row_sums_preview":["0","4","70","750","6500","50000","356250","2406250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D","basis A013611"],"edges":["e-7e9b074076a3cdbe","e-a9324a33426fa63e"],"text":"A131431 \u2192 D \u2192 basis A013611"},"headline":"A131431 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-bd4fd7073ba70c51","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-f9f48961069e0a5e","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-f9f48961069e0a5e","other":{"id":"n-f9f48961069e0a5e","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16"],["9","72","144"]],"card_rows":[["1"],["4","16"],["9","72","144"],["16","192","768","1024"]],"row_sums_preview":["1","20","225","2000","15625","112500","765625","5000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["(\u03b8+1)\u00b2","basis A013611"],"edges":["e-3411eba2e020f0d3","e-c6a4775a3ec17ab3"],"text":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611"},"headline":"basis-map coefficients of A026729 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611","chain":null}},{"id":"l-bdd20c9acf1772b4","kind":"fingerprint_related","node_a_id":"n-ef5c9478e7d83dab","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-ef5c9478e7d83dab","other":{"id":"n-ef5c9478e7d83dab","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","2"]],"card_rows":[["1"],["8","2"],["48","24","3"],["256","192","48","4"]],"row_sums_preview":["0","1","10","75","500","3125","18750","109375"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D"],"edges":["e-62aaaa59f91d42c1"],"text":"A038231 \u2192 D"},"headline":"A038231 \u2192 D","chain":null}},{"id":"l-be03677631070278","kind":"fingerprint_related","node_a_id":"n-c96fe4c1be387a19","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-c96fe4c1be387a19","other":{"id":"n-c96fe4c1be387a19","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["4","32","64"]],"card_rows":[["1"],["2","8"],["4","32","64"],["8","96","384","512"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611"],"edges":["e-d7d2422771947afc"],"text":"A130123 \u2192 basis A013611"},"headline":"A130123 \u2192 basis A013611","chain":null}},{"id":"l-c329099b52d04ba1","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-41592f90ceca7e00","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-41592f90ceca7e00","other":{"id":"n-41592f90ceca7e00","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-16"],["9","72","144"]],"card_rows":[["1"],["-4","-16"],["9","72","144"],["-16","-192","-768","-1024"]],"row_sums_preview":["1","-20","225","-2000","15625","-112500","765625","-5000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["(\u03b8+1)\u00b2","basis A013611"],"edges":["e-d58e809d3242b4fa","e-67015bcb57b73bc6"],"text":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611"},"headline":"basis-map coefficients of A030528, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611","chain":null}},{"id":"l-c44f24bee4d56bfa","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-0cc3f39711d63ef9","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-0cc3f39711d63ef9","other":{"id":"n-0cc3f39711d63ef9","name":"basis_map_A013611 of derivative of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","16"]],"card_rows":[["1"],["4","16"],["18","144","288"],["96","1152","4608","6144"]],"row_sums_preview":["0","1","20","450","12000","375000","13500000","551250000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["D","basis A013611"],"edges":["e-789a32df93c7cdb5","e-b3cf2c801874c157"],"text":"A136572 \u2192 D \u2192 basis A013611"},"headline":"A136572 \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-cbe27c7a600ff697","kind":"fingerprint_related","node_a_id":"n-21b46279e9abe0a4","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-21b46279e9abe0a4","other":{"id":"n-21b46279e9abe0a4","name":"basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","4"],["256","128","16"]],"card_rows":[["1"],["16","4"],["256","128","16"],["4096","3072","768","64"]],"row_sums_preview":["1","20","400","8000","160000","3200000","64000000","1280000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611"],"edges":["e-9c08db68889738c4"],"text":"A027467 \u2192 basis A013611"},"headline":"A027467 \u2192 basis A013611","chain":null}},{"id":"l-cefa9d6b43370a07","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-06c71d519c191976","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-06c71d519c191976","other":{"id":"n-06c71d519c191976","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["2","8"]],"card_rows":[["-1"],["2","8"],["-3","-24","-48"],["4","48","192","256"]],"row_sums_preview":["0","-1","10","-75","500","-3125","18750","-109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["D","basis A013611"],"edges":["e-7b1079618314be79","e-238349c351605a76"],"text":"basis-map coefficients of A030528, shifted \u2192 D \u2192 basis A013611"},"headline":"basis-map coefficients of A030528, shifted \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-d655b5ea253992c6","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-53d1470df2442b78","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-53d1470df2442b78","other":{"id":"n-53d1470df2442b78","name":"basis_map_A013611 of theta_plus_one_squared of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16"],["18","144","288"]],"card_rows":[["1"],["4","16"],["18","144","288"],["96","1152","4608","6144"]],"row_sums_preview":["1","20","450","12000","375000","13500000","551250000","25200000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["(\u03b8+1)\u00b2","basis A013611"],"edges":["e-76872140ee232c50","e-4aaf1b5e4e07af94"],"text":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611"},"headline":"A136572 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611","chain":null}},{"id":"l-d962daa099b43f51","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-f7c069e2c825f8ee","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-f7c069e2c825f8ee","other":{"id":"n-f7c069e2c825f8ee","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["16","128","256"],["64","768","3072","4096"],["256","4096","24576","65536","65536"]],"card_rows":[["16","128","256"],["64","768","3072","4096"],["256","4096","24576","65536","65536"],["1024","20480","163840","655360","1310720","1048576"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","reverse"],"edges":["e-7b68ca495e98f7ea","e-96ba13e2faa1ac85"],"text":"A038330 \u2192 basis A007318 \u2192 reverse"},"headline":"A038330 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-ded0e7fa8cade803","kind":"fingerprint_related","node_a_id":"n-8d4f9cf28f37366a","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-8d4f9cf28f37366a","other":{"id":"n-8d4f9cf28f37366a","name":"derivative of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["8","32"]],"card_rows":[["4"],["8","32"],["12","96","192"],["16","192","768","1024"]],"row_sums_preview":["0","4","40","300","2000","12500","75000","437500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["D"],"edges":["e-b04b02754d5cebbf"],"text":"A013611 \u2192 D"},"headline":"A013611 \u2192 D","chain":null}},{"id":"l-f61ee661b3884c30","kind":"fingerprint_related","node_a_id":"n-a4775661709ca859","node_b_id":"n-3ca3059e0c8c21b8","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-a4775661709ca859","other":{"id":"n-a4775661709ca859","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","2"],["64","32","4"]],"card_rows":[["1"],["8","2"],["64","32","4"],["512","384","96","8"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013609"],"edges":["e-13f16c9d95bb60f6"],"text":"A027466 \u2192 basis A013609"},"headline":"A027466 \u2192 basis A013609","chain":null}},{"id":"l-f626d680015d23fc","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-122589e065672009","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-122589e065672009","other":{"id":"n-122589e065672009","name":"basis_map_A013611 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":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["2","8"]],"card_rows":[["-1"],["2","8"],["-3","-24","-48"],["4","48","192","256"]],"row_sums_preview":["0","-1","10","-75","500","-3125","18750","-109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["D","basis A013611"],"edges":["e-c22d2d4fa11fd8cc","e-786460fbbc9178c5"],"text":"basis-map coefficients of A026729, shifted \u2192 D \u2192 basis A013611"},"headline":"basis-map coefficients of A026729, shifted \u2192 D \u2192 basis A013611","chain":null}},{"id":"l-f8cac10dbbf73c93","kind":"fingerprint_related","node_a_id":"n-3ca3059e0c8c21b8","node_b_id":"n-7dec8810ced708b9","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-7dec8810ced708b9","other":{"id":"n-7dec8810ced708b9","name":"basis_map_A013611 of theta_plus_one of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","32"],["27","216","432"]],"card_rows":[["1"],["8","32"],["27","216","432"],["64","768","3072","4096"]],"row_sums_preview":["1","40","675","8000","78125","675000","5359375","40000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["(\u03b8+1)\u00b2","basis A013611"],"edges":["e-048aefa516697baf","e-33cf796307aa0c88"],"text":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611"},"headline":"image under \u03b8+1 \u2192 (\u03b8+1)\u00b2 \u2192 basis A013611","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:            0;\n  1:            4;\n  2:          128,           32;\n  3:         3072,         1536,          192;\n  4:        65536,        49152,        12288,         1024;\n  5:      1310720,      1310720,       491520,        81920,        5120;\n  6:     25165824,     31457280,     15728640,      3932160,      491520,      24576;\n  7:    469762048,    704643072,    440401920,    146800640,    27525120,    2752512,     114688;\n  8:   8589934592,  15032385536,  11274289152,   4697620480,  1174405120,  176160768,   14680064,   524288;\n  9: 154618822656, 309237645312, 270582939648, 135291469824, 42278584320, 8455716864, 1056964608, 75497472, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (16*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n + 4*n*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((16*n + 4*n*x)*v[i - 1])/(n - 1)); 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-ff0214c333bfdcca","steps":["basis A013611","D"],"edges":["e-9c08db68889738c4","e-76ecb3d17f218949"],"text":"A027467 \u2192 basis A013611 \u2192 D"},"headline":"A027467 \u2192 basis A013611 \u2192 D","chain":null}
