{"id":"n-8d887bfc40d036b9","revision":3,"first_index":0,"fingerprint":"f6-f9762973ec1089d7","name":"Cube of lower triangular normalized binomial matrix.","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"],["3","1"],["9","6","1"],["27","27","9","1"],["81","108","54","12","1"],["243","405","270","90","15","1"],["729","1458","1215","540","135","18","1"],["2187","5103","5103","2835","945","189","21","1"],["6561","17496","20412","13608","5670","1512","252","24","1"],["19683","59049","78732","61236","30618","10206","2268","324","27","1"],["59049","196830","295245","262440","153090","61236","17010","3240","405","30","1"],["177147","649539","1082565","1082565","721710","336798","112266","26730","4455","495","33","1"],["531441","2125764","3897234","4330260","3247695","1732104","673596","192456","40095","5940","594","36","1"],["1594323","6908733","13817466","16888014","14073345","8444007","3752892","1250964","312741","57915","7722","702","39","1"],["4782969","22320522","48361131","64481508","59108049","39405366","19702683","7505784","2189187","486486","81081","9828","819","42","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","16","64","256","1024","4096","16384","65536","262144","1048576","4194304","16777216","67108864","268435456"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A027465","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158346,"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":false,"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-2786234a932620b2"]},{"id":158347,"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-090c65763405a203","e-2786234a932620b2","e-4b09f87a5a93f62b","e-58678fea8a24dd74","e-7b97c40e19745577"]},{"id":158348,"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-090c65763405a203","e-2786234a932620b2","e-4b09f87a5a93f62b","e-58678fea8a24dd74","e-7b97c40e19745577"]},{"id":158349,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-2786234a932620b2"]},{"id":158350,"position":4,"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-090c65763405a203","e-58678fea8a24dd74","e-7b97c40e19745577"]},{"id":158351,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)","region_exact":null,"edges":["e-72ff8ab34d4573c3"]},{"id":158352,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 3]","output_interval":"(-inf, 0]","statement":"an input root r <= 3 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 3] to roots in (-inf, 0]: an input root r <= 3 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-72ff8ab34d4573c3"]},{"id":158353,"position":7,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-4b09f87a5a93f62b"]},{"id":158354,"position":8,"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-4b09f87a5a93f62b"]}],"region_exact":{},"incoming_edges":[{"id":"e-7b97c40e19745577","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-8d887bfc40d036b9","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"ca4413ff99fe94b834b0ccb0346cbdfd308f65a0ac6122141eabf56cdea0a24f","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-af754f07daf65b5e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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]"}]},{"id":"e-72ff8ab34d4573c3","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-8d887bfc40d036b9","premise_count":0,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e8f4974c976fd2727e47294a37f93fbe5a685406c17c750e7ecdd8f30e3e168a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 3]","output_interval":"(-inf, 0]","statement":"an input root r <= 3 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-58678fea8a24dd74","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-8d887bfc40d036b9","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"1201408f22bf428eb672ae86817049aa4b391ed164eb56a5b64bcccf1567d422","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-090c65763405a203","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-8d887bfc40d036b9","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"16ec521548951cbc776cf0761f867dbdf13bd96ada154b62eb7255fc014014a4","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-6a5af5e1168bb679","role":"sequence","name":"Triangle of coefficients in expansion of (1+3*x)^n.","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]"}]},{"id":"e-4b09f87a5a93f62b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8d887bfc40d036b9","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"0e2bc69de65d0b8e6edbae4a3e1cb571cdec307c1c668a4808f090a8918e8742","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-af754f07daf65b5e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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]"}]},{"id":"e-2786234a932620b2","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-8d887bfc40d036b9","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"7f15c73a96f929ce8462bfef9f545c8a2c20520475117624d785438dfbd9a266","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-56706c372e88ce2b","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-fff9dc1c6c2891b5","conclusion_name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i."},{"id":"e-6d94a10ae1f2ed2c","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-41442279a7af7481","conclusion_name":"basis_map_A008277 of Cube of lower triangular normalized binomial matrix."},{"id":"e-9f0b70d11cb8d495","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-3dcaebc50e901d48","conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j)."},{"id":"e-b3c62654af8090af","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-f5249e29b34d4267","conclusion_name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i."},{"id":"e-d33b5447346a6a36","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-e8c6285b75cf03a2","conclusion_name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+3*x)^n."},{"id":"e-30d7db2c4a1b8418","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-a202ff64d2ae7451","conclusion_name":"obreschkoff_combination of Cube of lower triangular normalized binomial matrix."},{"id":"e-30d7db2c4a1b8418","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-a202ff64d2ae7451","conclusion_name":"obreschkoff_combination of Cube of lower triangular normalized binomial matrix."},{"id":"e-32b8bc759b20a254","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-fae37939907544d7","conclusion_name":"theta_plus_one_squared of Cube of lower triangular normalized binomial matrix."},{"id":"e-4db64ce997502606","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-3dcaebc50e901d48","conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j)."},{"id":"e-4f6f9107655e1886","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-d541bc30537dd88d","conclusion_name":"one_plus_derivative of Cube of lower triangular normalized binomial matrix."},{"id":"e-5cfa5af84579ea97","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-1adc5eb4389bea9b","conclusion_name":"mobius_x_over_x_plus_one of Cube of lower triangular normalized binomial matrix."},{"id":"e-6602f7f73d905d9e","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-f5aba8978740eb12","conclusion_name":"derivative of Cube of lower triangular normalized binomial matrix."},{"id":"e-b03bb7746bf562de","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-6a5af5e1168bb679","conclusion_name":"Triangle of coefficients in expansion of (1+3*x)^n."},{"id":"e-daddcb0470a384f9","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-462b6db6e919230d","conclusion_name":"theta_plus_one of Cube of lower triangular normalized binomial matrix."},{"id":"e-59cb5e4e13a9546c","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-abf94b4038449932","conclusion_name":"veronese_section_2_1 of Cube of lower triangular normalized binomial matrix."},{"id":"e-5d982f8f22366510","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-081b3f700db2d097","conclusion_name":"veronese_section_2_0 of Cube of lower triangular normalized binomial matrix."}],"used_by_total":16,"links":[{"id":"l-08918beecde713e8","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-7a4a43fcd9fa4a51","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-7a4a43fcd9fa4a51","other":{"id":"n-7a4a43fcd9fa4a51","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","3"],["81","54","9"]],"card_rows":[["1"],["9","3"],["81","54","9"],["729","729","243","27"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013610"],"edges":["e-b4f43a8046f9cfa3"],"text":"A038279 \u2192 basis A013610"},"headline":"A038279 \u2192 basis A013610","chain":null}},{"id":"l-2a800a2bae1ef829","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-3e79e2d28a06cec2","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-3e79e2d28a06cec2","other":{"id":"n-3e79e2d28a06cec2","name":"basis_map_A013610 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-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["1","6","9"]],"card_rows":[["1"],["1","3"],["1","6","9"],["1","9","27","27"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013610"],"edges":["e-41e3c7da1a12c920"],"text":"basis-map coefficients of A026729 \u2192 basis A013610"},"headline":"basis-map coefficients of A026729 \u2192 basis A013610","chain":null}},{"id":"l-30c4843a47b5c017","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-a22bdff1819212bb","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-a22bdff1819212bb","other":{"id":"n-a22bdff1819212bb","name":"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-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","3"],["27","18","3"]],"card_rows":[["3"],["9","3"],["27","18","3"],["81","81","27","3"]],"row_sums_preview":["3","12","48","192","768","3072","12288","49152"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A007318"],"edges":["e-18f137fab6db112d"],"text":"A120910 \u2192 basis A007318"},"headline":"A120910 \u2192 basis A007318","chain":null}},{"id":"l-407e249b499a5a09","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-412b54b3d1f71393","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-412b54b3d1f71393","other":{"id":"n-412b54b3d1f71393","name":"basis_map_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["2","12","18"]],"card_rows":[["1"],["1","3"],["2","12","18"],["6","54","162","162"]],"row_sums_preview":["1","4","32","384","6144","122880","2949120","82575360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013610"],"edges":["e-77bf2cb365051314"],"text":"A136572 \u2192 basis A013610"},"headline":"A136572 \u2192 basis A013610","chain":null}},{"id":"l-408a1d2348cdaf1a","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-96b6c636cb527859","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-96b6c636cb527859","other":{"id":"n-96b6c636cb527859","name":"basis_map_A013610 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-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-3"],["1","6","9"]],"card_rows":[["1"],["-1","-3"],["1","6","9"],["-1","-9","-27","-27"]],"row_sums_preview":["1","-4","16","-64","256","-1024","4096","-16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013610"],"edges":["e-226103a223be7e17"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013610"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013610","chain":null}},{"id":"l-48e8b17924277a7f","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-a93042cd70c4761e","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-a93042cd70c4761e","other":{"id":"n-a93042cd70c4761e","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","96","16"],["1728","1728","576","64"],["20736","27648","13824","3072","256"]],"card_rows":[["144","96","16"],["1728","1728","576","64"],["20736","27648","13824","3072","256"],["248832","414720","276480","92160","15360","1024"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":[],"edges":[],"text":"A038330"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","chain":null}},{"id":"l-734e5428d253dacb","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-af427f3533ba7386","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-af427f3533ba7386","other":{"id":"n-af427f3533ba7386","name":"basis_map_A013610 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-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-3"],["2","12","18"]],"card_rows":[["1"],["-1","-3"],["2","12","18"],["-6","-54","-162","-162"]],"row_sums_preview":["1","-4","32","-384","6144","-122880","2949120","-82575360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A013610"],"edges":["e-46ba877ad14f98f3"],"text":"basis-map coefficients of A021012 \u2192 basis A013610"},"headline":"basis-map coefficients of A021012 \u2192 basis A013610","chain":null}},{"id":"l-767c3dc84a29c0ab","kind":"fingerprint_related","node_a_id":"n-6a5af5e1168bb679","node_b_id":"n-8d887bfc40d036b9","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-6a5af5e1168bb679","other":{"id":"n-6a5af5e1168bb679","name":"Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["1","6","9"]],"card_rows":[["1"],["1","3"],["1","6","9"],["1","9","27","27"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013610"],"derivation":{"root":"n-6a5af5e1168bb679","steps":[],"edges":[],"text":"A013610"},"headline":"A013610: Triangle of coefficients in expansion of (1+3*x)^n.","chain":null}},{"id":"l-8538051dab44db98","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-d4ffddfb9dca0c4b","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-d4ffddfb9dca0c4b","other":{"id":"n-d4ffddfb9dca0c4b","name":"basis_map_A013610 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-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","6"],["4","24","36"]],"card_rows":[["1"],["2","6"],["4","24","36"],["8","72","216","216"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013610"],"edges":["e-db1885975295b772"],"text":"A130123 \u2192 basis A013610"},"headline":"A130123 \u2192 basis A013610","chain":null}},{"id":"l-8c338e9f7fbc6db3","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-fe3df7f96c918e14","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-fe3df7f96c918e14","other":{"id":"n-fe3df7f96c918e14","name":"derivative of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["6","18"]],"card_rows":[["3"],["6","18"],["9","54","81"],["12","108","324","324"]],"row_sums_preview":["0","3","24","144","768","3840","18432","86016"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["D"],"edges":["e-5fa4095a1ef98485"],"text":"A013610 \u2192 D"},"headline":"A013610 \u2192 D","chain":null}},{"id":"l-8f53d7361a3fcd13","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-4575a0f9ca60b9ab","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-4575a0f9ca60b9ab","other":{"id":"n-4575a0f9ca60b9ab","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["3","9"],["3","18","27"]],"card_rows":[["3"],["3","9"],["3","18","27"],["3","27","81","81"]],"row_sums_preview":["3","12","48","192","768","3072","12288","49152"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["x \u21a6 x/(x+1)"],"edges":["e-e61ce0e3d93770a8"],"text":"A120909 \u2192 x \u21a6 x/(x+1)"},"headline":"A120909 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-9101ef173aeb1efa","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-1cfea80692e421ba","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-1cfea80692e421ba","other":{"id":"n-1cfea80692e421ba","name":"basis_map_A013610 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-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-3"],["1","6","9"]],"card_rows":[["1"],["-1","-3"],["1","6","9"],["-1","-9","-27","-27"]],"row_sums_preview":["1","-4","16","-64","256","-1024","4096","-16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013610"],"edges":["e-fe4cc1f9dd1d3096"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013610"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013610","chain":null}},{"id":"l-9b3f051f945b5934","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-f5aba8978740eb12","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-f5aba8978740eb12","other":{"id":"n-f5aba8978740eb12","name":"derivative of Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["27","18","3"],["108","108","36","4"]],"row_sums_preview":["0","1","8","48","256","1280","6144","28672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["D"],"edges":["e-6602f7f73d905d9e"],"text":"A027465 \u2192 D"},"headline":"A027465 \u2192 D","chain":null}},{"id":"l-a805e0bcc0ec6ede","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-63e0d27d213f2270","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-63e0d27d213f2270","other":{"id":"n-63e0d27d213f2270","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","12"],["16","96","144"]],"card_rows":[["1"],["4","12"],["16","96","144"],["64","576","1728","1728"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A013611"],"edges":["e-82b80af703a32f5e"],"text":"A013610 \u2192 basis A013611"},"headline":"A013610 \u2192 basis A013611","chain":null}},{"id":"l-d580e0851283d2c9","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-ff3307aeb79ab12b","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-ff3307aeb79ab12b","other":{"id":"n-ff3307aeb79ab12b","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","2"],["36","24","4"]],"card_rows":[["1"],["6","2"],["36","24","4"],["216","216","72","8"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013609"],"edges":["e-037f2552a98ec0cc"],"text":"A038243 \u2192 basis A013609"},"headline":"A038243 \u2192 basis A013609","chain":null}},{"id":"l-def7fc37807af55c","kind":"fingerprint_related","node_a_id":"n-8d887bfc40d036b9","node_b_id":"n-b8d8f9de0e27e805","fingerprint":"f6-f9762973ec1089d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f9762973ec1089d7"},"other_id":"n-b8d8f9de0e27e805","other":{"id":"n-b8d8f9de0e27e805","name":"basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-f9762973ec1089d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","6","9"],["1","9","27","27"],["1","12","54","108","81"]],"card_rows":[["1","6","9"],["1","9","27","27"],["1","12","54","108","81"],["1","15","90","270","405","243"]],"row_sums_preview":["16","64","256","1024","4096","16384","65536","262144"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013610"],"edges":["e-33def10999a9e794"],"text":"basis-map coefficients of A112227 \u2192 basis A013610"},"headline":"basis-map coefficients of A112227 \u2192 basis A013610","chain":null}},{"id":"l-02d260752601db7e","kind":"oeis_match","node_a_id":"n-8d887bfc40d036b9","node_b_id":null,"fingerprint":null,"oeis_id":"A347478","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A347478","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2840ae6b7512c5d8","kind":"oeis_match","node_a_id":"n-8d887bfc40d036b9","node_b_id":null,"fingerprint":null,"oeis_id":"A000302","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000302","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-602f34d9207b7914","kind":"oeis_match","node_a_id":"n-8d887bfc40d036b9","node_b_id":null,"fingerprint":null,"oeis_id":"A027465","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":52},"in_catalog":true,"oeis_id":"A027465","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ae86766015da09d1","kind":"oeis_match","node_a_id":"n-8d887bfc40d036b9","node_b_id":null,"fingerprint":null,"oeis_id":"A013610","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A013610","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fba55a1ec13a5a53","kind":"oeis_match","node_a_id":"n-8d887bfc40d036b9","node_b_id":null,"fingerprint":null,"oeis_id":"A155543","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A155543","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A027465","relation":"identification"},{"oeis_id":"A013610","relation":"match_reversed_triangle"},{"oeis_id":"A000302","relation":"match_row_sums"},{"oeis_id":"A155543","relation":"match_row_sums"},{"oeis_id":"A347478","relation":"match_row_sums"},{"oeis_id":"A027465","relation":"match_triangle"}],"bfiles":[{"node_id":"n-8d887bfc40d036b9","oeis_id":"A000302","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":12972},{"node_id":"n-8d887bfc40d036b9","oeis_id":"A013610","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":17411},{"node_id":"n-8d887bfc40d036b9","oeis_id":"A027465","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":52,"oeis_terms_agreeing":52,"oeis_terms_available":52,"term_count":990,"raw_size":17411}],"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:     3,     1;\n  2:     9,     6,     1;\n  3:    27,    27,     9,     1;\n  4:    81,   108,    54,    12,     1;\n  5:   243,   405,   270,    90,    15,     1;\n  6:   729,  1458,  1215,   540,   135,    18,    1;\n  7:  2187,  5103,  5103,  2835,   945,   189,   21,   1;\n  8:  6561, 17496, 20412, 13608,  5670,  1512,  252,  24,  1;\n  9: 19683, 59049, 78732, 61236, 30618, 10206, 2268, 324, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + x)*v[i - 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}},{"oeis_id":"A013610","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  3;\n  2: 1,  6,   9;\n  3: 1,  9,  27,   27;\n  4: 1, 12,  54,  108,    81;\n  5: 1, 15,  90,  270,   405,   243;\n  6: 1, 18, 135,  540,  1215,  1458,   729;\n  7: 1, 21, 189,  945,  2835,  5103,  5103,  2187;\n  8: 1, 24, 252, 1512,  5670, 13608, 20412, 17496,  6561;\n  9: 1, 27, 324, 2268, 10206, 30618, 61236, 78732, 59049, 19683;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A027465","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     3,     1;\n  2:     9,     6,     1;\n  3:    27,    27,     9,     1;\n  4:    81,   108,    54,    12,     1;\n  5:   243,   405,   270,    90,    15,     1;\n  6:   729,  1458,  1215,   540,   135,    18,    1;\n  7:  2187,  5103,  5103,  2835,   945,   189,   21,   1;\n  8:  6561, 17496, 20412, 13608,  5670,  1512,  252,  24,  1;\n  9: 19683, 59049, 78732, 61236, 30618, 10206, 2268, 324, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-8d887bfc40d036b9","steps":[],"edges":[],"text":"A027465"},"headline":"A027465: Cube of lower triangular normalized binomial matrix.","chain":null}