{"id":"n-8afe6c02ab69bf6d","revision":1,"first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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":[["196","224","64"],["2744","4704","2688","512"],["38416","87808","75264","28672","4096"],["537824","1536640","1756160","1003520","286720","32768"],["7529536","25815552","36879360","28098560","12042240","2752512","262144"],["105413504","421654016","722835456","688414720","393379840","134873088","25690112","2097152"],["1475789056","6746464256","13492928512","15420489728","11014635520","5035261952","1438646272","234881024","16777216"],["20661046784","106256812032","242872713216","323830284288","277568815104","158610751488","60423143424","14797504512","2113929216","134217728"],["289254654976","1652883742720","4250272481280","6476605685760","6476605685760","4441101041664","2114810019840","690550210560","147975045120","18790481920","1073741824"],["4049565169664","25454409637888","72726884679680","124674659450880","142485325086720","113988260069376","65136148611072","26586183106560","7596052316160","1446867107840","165356240896","8589934592"],["56693912375296","388758256287744","1221811662618624","2327260309749760","2992191826821120","2735718241665024","1823812161110016","893295752380416","319034197278720","81024558039040","13889924235264","1443109011456","68719476736"],["793714773254144","5896166887030784","20215429326962688","42356137637445632","60508768053493760","62237589997879296","47419116188860416","27096637822205952","11612844780945408","3686617390776320","842655403606016","131322920042496","12506944765952","549755813888"],["11112006825558016","88896054604464128","330185345673723904","754709361539940352","1185971853848477696","1355396404398260224","1161768346627080192","758705859021766656","379352929510883328","144515401718431744","41290114776694784","8579764109443072","1225680587063296","107752139522048","4398046511104"],["155568095557812224","1333440819066961920","5333763276267847680","13207413826948956160","22641280846198210560","28463324492363464704","27107928087965204480","19916028799321374720","11380587885326499840","5058039060145111040","1734184820621180928","450437615745761280","85797641094430720","11313974649815040","923589767331840","35184372088832"],["2177953337809371136","19912716231399964672","85340212420285562880","227573899787428167680","422637242462366597120","579616789662674190336","607217589170420580352","495687827894220881920","318656460789141995520","161857249924643553280","64742899969857421312","20179605185410105344","4804667901288120320","844776773852856320","103442053941166080","7881299347898368","281474976710656"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792","26559922791424","584318301411328","12855002631049216","282810057883082752","6221821273427820544","136880068015412051968","3011361496339065143296"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["196","224","64"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["14","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168249,"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-0eb3975e7d8d74fc"]},{"id":168250,"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-0eb3975e7d8d74fc"]},{"id":168251,"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-0eb3975e7d8d74fc"]}],"region_exact":{},"incoming_edges":[{"id":"e-0eb3975e7d8d74fc","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-8afe6c02ab69bf6d","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":"8b938124bdedf59ad566e1e19b9576cc94d2d40fd198648f2579a78dcf966374","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-ea9b82d727693fb6","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-efd812fee87252af","kind":"fingerprint_related","node_a_id":"n-ade19c5257c8a726","node_b_id":"n-8afe6c02ab69bf6d","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-ade19c5257c8a726","other":{"id":"n-ade19c5257c8a726","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","4"],["49","56","16"]],"card_rows":[["1"],["7","4"],["49","56","16"],["343","588","336","64"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013611"],"edges":["e-1b27bff9bafb5517"],"text":"A038255 \u2192 basis A013611"},"headline":"A038255 \u2192 basis A013611","chain":null}},{"id":"l-a384c0c368606052","kind":"oeis_match","node_a_id":"n-8afe6c02ab69bf6d","node_b_id":null,"fingerprint":null,"oeis_id":"A009966","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":"A009966","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009966","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:           196,            224,             64;\n  1:          2744,           4704,           2688,             512;\n  2:         38416,          87808,          75264,           28672,            4096;\n  3:        537824,        1536640,        1756160,         1003520,          286720,           32768;\n  4:       7529536,       25815552,       36879360,        28098560,        12042240,         2752512,         262144;\n  5:     105413504,      421654016,      722835456,       688414720,       393379840,       134873088,       25690112,        2097152;\n  6:    1475789056,     6746464256,    13492928512,     15420489728,     11014635520,      5035261952,     1438646272,      234881024,      16777216;\n  7:   20661046784,   106256812032,   242872713216,    323830284288,    277568815104,    158610751488,    60423143424,    14797504512,    2113929216,     134217728;\n  8:  289254654976,  1652883742720,  4250272481280,   6476605685760,   6476605685760,   4441101041664,  2114810019840,   690550210560,  147975045120,   18790481920,   1073741824;\n  9: 4049565169664, 25454409637888, 72726884679680, 124674659450880, 142485325086720, 113988260069376, 65136148611072, 26586183106560, 7596052316160, 1446867107840, 165356240896, 8589934592;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (14+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 224*x + 64*x^2.","mathematica":"P[0, x_] := 196 + 224*x + 64*x^2;\nP[n_, x_] := P[n, x] = Expand[(14 + 8*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] = 196 + 224*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 8*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-ea9b82d727693fb6","steps":["basis A013611"],"edges":["e-0eb3975e7d8d74fc"],"text":"A038328 \u2192 basis A013611"},"headline":"A038328 \u2192 basis A013611","chain":null}
