{"id":"n-9a60224d294916c0","revision":0,"first_index":1,"fingerprint":"f6-40f5d787e72f16e0","name":"shift_x_plus_one of basis_map_A013610 of Coefficients of a special case of Poisson-Charlier polynomials.","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":false,"rows":[["1"],["5","3"],["29","33","9"],["217","354","180","27"],["2297","4950","3717","1080","81"],["34349","94917","98991","46683","8829","243"],["674965","2319567","3188556","2193723","757674","106677","729"],["16276481","67383420","116153001","106715853","55125279","15189201","1749600","2187"],["461527793","2241102780","4660922790","5381788068","3726030699","1546808724","356540778","35219448","6561"],["14993138773","83631891159","203986810890","284158462230","247262918139","137622938949","47846548854","9499894974","824776749","19683"],["548258687501","3456292897101","9679781147787","15806835148518","16585984121265","11596940120331","5403135628449","1617516244350","282324547359","21889917009","59049"],["22272738733865","156646261166118","495588891144618","928791300785571","1141875934540479","962262749230740","562901511454218","225702487308357","59364512224245","9248816211552","648135287172","177147"],["994870668959209","7724437548168882","27252740651158755","57672335613731292","81337971602581722","80273770406868006","56569076543183679","28464668553632436","10022526806028822","2351794020351858","330986934015759","21165761645568","531441"],["48451779617935997","411704515243112121","1602975672791134353","3781515346316529561","6019877909570716494","6812766546579889776","5620306739748552513","3405446528747743341","1504122328382275170","472275007651269432","100062805932275229","12844740646264329","755466070310613","1594323"],["2554818339078836357","23585049454076302107","100466208550567427526","261468892913316428313","463856047929818346105","592340510705583206304","560165352156595779255","397201101129828164745","211179482805255099561","83146245487688740824","23563889606562574455","4551957763776264033","537211484127006012","29253180765581109","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","71","778","12125","284012","9241891","378595022","18409947641","1029827400400","64998958518719","4565303338264082","353016345110857429","29793105387299603252","2724646021507044539675"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["5","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["32","48","18"],["-16","-24","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","-3"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187663,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-a9400587536c434c"]},{"id":187664,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-a9400587536c434c","e-b902b3985b593494"]},{"id":187665,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-a9400587536c434c","e-b902b3985b593494"]},{"id":187666,"position":3,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b902b3985b593494"]},{"id":187667,"position":4,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-b902b3985b593494"]}],"region_exact":{},"incoming_edges":[{"id":"e-b902b3985b593494","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9a60224d294916c0","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":"f2e54c79ab03aa2102c79a96f56327679be3f1dfd9279bd1f92dd9e17e6fa9fb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6c2910cdb87c65c2","role":"sequence","name":"basis_map_A013610 of Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-a9400587536c434c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9a60224d294916c0","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":"09d7075e3c3fbd8c1834ee283d0cd99f1a0e08d875847d720c09e0c5272df757","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-6c2910cdb87c65c2","role":"sequence","name":"basis_map_A013610 of Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:           1;\n   2:           5,           3;\n   3:          29,          33,            9;\n   4:         217,         354,          180,           27;\n   5:        2297,        4950,         3717,         1080,           81;\n   6:       34349,       94917,        98991,        46683,         8829,          243;\n   7:      674965,     2319567,      3188556,      2193723,       757674,       106677,         729;\n   8:    16276481,    67383420,    116153001,    106715853,     55125279,     15189201,     1749600,       2187;\n   9:   461527793,  2241102780,   4660922790,   5381788068,   3726030699,   1546808724,   356540778,   35219448,      6561;\n  10: 14993138773, 83631891159, 203986810890, 284158462230, 247262918139, 137622938949, 47846548854, 9499894974, 824776749, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n-32+(24*n-48)*x+(9*n-18)*x^2)*P(n-2,x) + (4*n-3+(3*n-3)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n - 32 + (24*n - 48)*x + (9*n - 18)*x^2)*P[n - 2, x] + (4*n - 3 + (3*n - 3)*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = -(16*n - 32 + (24*n - 48)*x + (9*n - 18)*x^2)*v[i - 2] + (4*n - 3 + (3*n - 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-3699a0e7eb346ca6","e-a9400587536c434c"],"text":"A046716 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A046716 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}