{"id":"n-b3aee49351d9bfdc","revision":0,"first_index":0,"fingerprint":"f6-7e79d1acae7aa409","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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"],["7","4"],["70","82","24"],["910","1626","968","192"],["14560","35116","31748","12752","1920"],["276640","841924","1024604","623264","189504","23040"],["6086080","22395288","34328224","28056264","12894784","3159936","322560"],["152152000","657259480","1216530208","1250658184","771269824","285314944","58622976","5160960"],["4260256000","21142001440","45893516464","56915972896","44107402384","21871675264","6777112320","1199720448","92897280"],["132067936000","740607164640","1845539039184","2682265489056","2505648931824","1560169980864","647523987200","172733580288","26874224640","1857945600"],["4490309824000","28086138189760","79041684954336","131798885489952","144201904441248","108170055849504","56339555143808","20118469448192","4713862404096","654403092480","40874803200"],["166141463488000","1146954548797120","3598609659864672","6773559202032288","8498643716085024","7463137773021600","4680644880708992","2096532693034496","657256175708160","137345612120064","17218041937920","980995276800"],["6645658539520000","50197860002572800","173765204663312000","364506219237772992","516058287896240448","519490247539074624","381267377326921280","205558074619813632","80800097047223296","22582485053214720","4259707592638464","486908901457920","25505877196800"],["285763317199360000","2344586419217190400","8877443880594454400","20539193157796974656","32396680518195983040","36787712705274941376","30940224156151704512","19514483773805782016","9230030262385383424","3233449574610485248","815477007973466112","140208895356567552","14730201960284160","714164561510400"],["13145112591170560000","116423874799971558400","478700011083860614400","1211126201676494466176","2106423098570924459520","2664135199988526794496","2526881692341226648832","1825872978279617108096","1010015905283901097984","425639588303643824128","134515429605093998592","30913919425606090752","4883856150870097920","474757628638003200","21424936845312000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","176","3696","96096","2978976","107243136","4396968576","202260554496","10315288279296","577656143640576","35237024762075136","2325643634296958976","165120698035084087296","12549173050666390634496"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","2"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189186,"position":0,"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-5cc2c38f0be4ab33"]},{"id":189187,"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-25333412e4d508f4","e-5cc2c38f0be4ab33"]},{"id":189188,"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-25333412e4d508f4","e-5cc2c38f0be4ab33"]},{"id":189189,"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-25333412e4d508f4","e-5cc2c38f0be4ab33"]},{"id":189190,"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-25333412e4d508f4"]}],"region_exact":{},"incoming_edges":[{"id":"e-25333412e4d508f4","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-b3aee49351d9bfdc","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":"62b8037a7cea3dd51294c40d0bc792874c25b91ea9ea7e60c49c0176f436489f","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-7ce10652a7d44cf0","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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-5cc2c38f0be4ab33","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b3aee49351d9bfdc","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":"0c5af64a1db5e60ab32adc1d3e99fe2cbe2fb015c171deb3253b222b2d9ff01d","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-7ce10652a7d44cf0","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:            1;\n  1:            7,            4;\n  2:           70,           82,            24;\n  3:          910,         1626,           968,           192;\n  4:        14560,        35116,         31748,         12752,          1920;\n  5:       276640,       841924,       1024604,        623264,        189504,         23040;\n  6:      6086080,     22395288,      34328224,      28056264,      12894784,       3159936,       322560;\n  7:    152152000,    657259480,    1216530208,    1250658184,     771269824,     285314944,     58622976,      5160960;\n  8:   4260256000,  21142001440,   45893516464,   56915972896,   44107402384,   21871675264,   6777112320,   1199720448,    92897280;\n  9: 132067936000, 740607164640, 1845539039184, 2682265489056, 2505648931824, 1560169980864, 647523987200, 172733580288, 26874224640, 1857945600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+4+(2*n+2)*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*n + 4 + (2*n + 2)*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*n + 4 + (2*n + 2)*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-2feb6dc01f424b0a","steps":["basis A013609","basis A007318"],"edges":["e-5f6e941a4a66ef73","e-25333412e4d508f4"],"text":"A145324 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A145324 \u2192 basis A013609 \u2192 basis A007318","chain":null}
