{"id":"n-108b34f10b7468f4","revision":1,"first_index":0,"fingerprint":"f6-10cba46fcb583bff","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["64","48","9"],["512","576","216","27"],["4096","6144","3456","864","81"],["32768","61440","46080","17280","3240","243"],["262144","589824","552960","276480","77760","11664","729"],["2097152","5505024","6193152","3870720","1451520","326592","40824","2187"],["16777216","50331648","66060288","49545216","23224320","6967296","1306368","139968","6561"],["134217728","452984832","679477248","594542592","334430208","125411328","31352832","5038848","472392","19683"],["1073741824","4026531840","6794772480","6794772480","4459069440","2006581248","627056640","134369280","18895680","1574640","59049"],["8589934592","35433480192","66437775360","74742497280","56056872960","29429858304","11036196864","2956124160","554273280","69284160","5196312","177147"],["68719476736","309237645312","637802643456","797253304320","672682475520","403609485312","176579149824","56757583872","13302558720","2217093120","249422976","17006112","531441"],["549755813888","2680059592704","6030134083584","8291434364928","7773219717120","5246923309056","2623461654528","983798120448","276693221376","57644421120","8646663168","884317824","55269864","1594323"],["4398046511104","23089744183296","56281251446784","84421877170176","87060060831744","65295045623808","36728463163392","15740769927168","5164940132352","1291235033088","242106568704","33014532096","3095112384","178564176","4782969"],["35184372088832","197912092999680","519519244124160","844218771701760","949746118164480","783540547485696","489712842178560","236111548907520","88541830840320","25824700661760","5810557648896","990435962880","123804495360","10713850560","573956280","14348907"],["281474976710656","1688849860263936","4749890231992320","8312307905986560","10130625260421120","9117562734379008","6268324379885568","3358030917795840","1416669293445120","472223097815040","123958563176448","25355160649728","3961743851520","457124290560","36733201920","1836660096","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["121","1331","14641","161051","1771561","19487171","214358881","2357947691","25937424601","285311670611","3138428376721","34522712143931","379749833583241","4177248169415651","45949729863572161"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A038281","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["64","48","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158509,"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-da334b7c58401d78"]},{"id":158510,"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-c5656743efdffc45","e-da334b7c58401d78"]},{"id":158511,"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-c5656743efdffc45","e-da334b7c58401d78"]},{"id":158512,"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-da334b7c58401d78"]},{"id":158513,"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-c5656743efdffc45"]},{"id":158514,"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-a1af237376376fe3"]},{"id":158515,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 8]","output_interval":"(-inf, 0]","statement":"an input root r <= 8 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, 8] to roots in (-inf, 0]: an input root r <= 8 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-a1af237376376fe3"]}],"region_exact":{},"incoming_edges":[{"id":"e-a1af237376376fe3","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-108b34f10b7468f4","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":"0e119d5b67c8f0b10d5cd5aa639b1fa91d8104480fa46150295d80585145ac9a","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, 8]","output_interval":"(-inf, 0]","statement":"an input root r <= 8 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-c5656743efdffc45","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-108b34f10b7468f4","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":"0f3a55fd6bec37727664bb0c541ec56e30abe3c1c6ba6e877df7669fb314b1d2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-da334b7c58401d78","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-108b34f10b7468f4","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":"37b0cbd76b4f98fd4029f4f9936b9a4696eb8e9ed99ade6d7cfa0bd24affeb9f","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-2fddb004b6f97d1c","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-1f2dec04776afb2b","conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j."},{"id":"e-8f18b93eb5dfe7b3","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-f55a7ba0b32633bc","conclusion_name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j."},{"id":"e-d7f363249f6694ec","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-db2ea4f508716e1f","conclusion_name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j."},{"id":"e-e643036831fc9f95","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-809ca598570cd420","conclusion_name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j."},{"id":"e-f67f09ded1c0d580","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-4113d906b9d95b00","conclusion_name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j."},{"id":"e-b64329fdd6480cca","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-15f7f9eaa70b650d","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j."},{"id":"e-b64329fdd6480cca","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-15f7f9eaa70b650d","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j."}],"used_by_total":7,"links":[{"id":"l-4ef519e65bb89119","kind":"fingerprint_related","node_a_id":"n-108b34f10b7468f4","node_b_id":"n-5480ea27b6b60650","fingerprint":"f6-10cba46fcb583bff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-10cba46fcb583bff"},"other_id":"n-5480ea27b6b60650","other":{"id":"n-5480ea27b6b60650","name":"basis_map_A013611 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":"f6-10cba46fcb583bff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","24"],["27","144","192"]],"card_rows":[["3"],["9","24"],["27","144","192"],["81","648","1728","1536"]],"row_sums_preview":["3","33","363","3993","43923","483153","5314683","58461513"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013611"],"edges":["e-2f246bda2100be8f"],"text":"A120909 \u2192 basis A013611"},"headline":"A120909 \u2192 basis A013611","chain":null}},{"id":"l-55c6159c9b71fabb","kind":"fingerprint_related","node_a_id":"n-26633b980c3508e8","node_b_id":"n-108b34f10b7468f4","fingerprint":"f6-10cba46fcb583bff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-10cba46fcb583bff"},"other_id":"n-26633b980c3508e8","other":{"id":"n-26633b980c3508e8","name":"Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-10cba46fcb583bff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","48","64"],["27","216","576","512"],["81","864","3456","6144","4096"]],"card_rows":[["9","48","64"],["27","216","576","512"],["81","864","3456","6144","4096"],["243","3240","17280","46080","61440","32768"]],"row_sums_preview":["121","1331","14641","161051","1771561","19487171","214358881","2357947691"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":[],"edges":[],"text":"A038226"},"headline":"Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","chain":null}},{"id":"l-8adc313c0c085fb9","kind":"fingerprint_related","node_a_id":"n-108b34f10b7468f4","node_b_id":"n-9df1aeebd1dc8cfe","fingerprint":"f6-10cba46fcb583bff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-10cba46fcb583bff"},"other_id":"n-9df1aeebd1dc8cfe","other":{"id":"n-9df1aeebd1dc8cfe","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-10cba46fcb583bff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","8"],["9","48","64"]],"card_rows":[["1"],["3","8"],["9","48","64"],["27","216","576","512"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013611"],"edges":["e-4b355511bff796f6"],"text":"A013609 \u2192 basis A013611"},"headline":"A013609 \u2192 basis A013611","chain":null}},{"id":"l-c44564482998f158","kind":"fingerprint_related","node_a_id":"n-108b34f10b7468f4","node_b_id":"n-a227586ec36efe06","fingerprint":"f6-10cba46fcb583bff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-10cba46fcb583bff"},"other_id":"n-a227586ec36efe06","other":{"id":"n-a227586ec36efe06","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-10cba46fcb583bff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","3"],["64","48","9"]],"card_rows":[["1"],["8","3"],["64","48","9"],["512","576","216","27"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013610"],"edges":["e-892638ce5a781886"],"text":"A027466 \u2192 basis A013610"},"headline":"A027466 \u2192 basis A013610","chain":null}},{"id":"l-ffaba634ff097398","kind":"oeis_match","node_a_id":"n-108b34f10b7468f4","node_b_id":null,"fingerprint":null,"oeis_id":"A001020","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001020","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038281","relation":"identification"},{"oeis_id":"A001020","relation":"match_row_sums"}],"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:         64,          48,           9;\n  1:        512,         576,         216,          27;\n  2:       4096,        6144,        3456,         864,          81;\n  3:      32768,       61440,       46080,       17280,        3240,         243;\n  4:     262144,      589824,      552960,      276480,       77760,       11664,         729;\n  5:    2097152,     5505024,     6193152,     3870720,     1451520,      326592,       40824,       2187;\n  6:   16777216,    50331648,    66060288,    49545216,    23224320,     6967296,     1306368,     139968,      6561;\n  7:  134217728,   452984832,   679477248,   594542592,   334430208,   125411328,    31352832,    5038848,    472392,    19683;\n  8: 1073741824,  4026531840,  6794772480,  6794772480,  4459069440,  2006581248,   627056640,  134369280,  18895680,  1574640,   59049;\n  9: 8589934592, 35433480192, 66437775360, 74742497280, 56056872960, 29429858304, 11036196864, 2956124160, 554273280, 69284160, 5196312, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 48*x + 9*x^2.","mathematica":"P[0, x_] := 64 + 48*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(8 + 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] = 64 + 48*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + 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}}],"derivation":{"root":"n-108b34f10b7468f4","steps":[],"edges":[],"text":"A038281"},"headline":"A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","chain":null}