{"id":"n-3f3ffed71cecca6c","revision":1,"first_index":0,"fingerprint":"f6-50110cf0be5eacc9","name":"basis_map_A013610 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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":false,"rows":[["1"],["5","12"],["17","72","72"],["61","372","720","432"],["217","1728","4968","6048","2592"],["773","7596","28944","53136","46656","15552"],["2753","32184","152712","374976","500256","342144","93312"],["9805","132900","754704","2321136","4162752","4339008","2426112","559872"],["34921","538416","3560040","13157856","29665440","41679360","35551872","16796160","3359232"],["124373","2149404","16217424","70004304","190195776","336685248","387618048","279376128","114213888","20155392"],["442961","8480808","71899272","354993408","1128891168","2412581760","3504145536","3410740224","2126393856","765904896","120932352"],["1577629","33145236","311913936","1733689008","6317051328","15825637440","27783274752","34141554432","28748307456","15781671936","5079158784","725594112"],["5618809","128520864","1329296616","8213933088","33741030816","96977730816","199787430528","296970665472","315811478016","234125033472","114764802048","33377329152","4353564672"],["20011685","495031692","5581714320","37956440592","173542927680","562932960192","1331749050624","2324619336960","2991604368384","2801297157120","1853892956160","820646940672","217678233600","26121388032"],["71272673","1895919768","23145051528","171763253568","865133086752","3126458093184","8354286158976","16757018228736","25335976075776","28716636616704","24003741616128","14340642029568","5785887449088","1410554953728","156728328192"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","17","161","1585","15553","152657","1498337","14706289","144343297","1416739985","13905406241","136482576049","1339586434369","13148138517713","129049938134369"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167229,"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-d7c15f9b4df77eea"]},{"id":167230,"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-d7c15f9b4df77eea"]},{"id":167231,"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-d7c15f9b4df77eea"]},{"id":167232,"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-d7c15f9b4df77eea"]}],"region_exact":{},"incoming_edges":[{"id":"e-d7c15f9b4df77eea","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-3f3ffed71cecca6c","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":"2d5dafe342aebb37079612ee0bf523daba977d6f04a9abaae7e7240e08e43bd9","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-e646c30d067149e7","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:      5,      12;\n  2:     17,      72,       72;\n  3:     61,     372,      720,      432;\n  4:    217,    1728,     4968,     6048,      2592;\n  5:    773,    7596,    28944,    53136,     46656,     15552;\n  6:   2753,   32184,   152712,   374976,    500256,    342144,     93312;\n  7:   9805,  132900,   754704,  2321136,   4162752,   4339008,   2426112,    559872;\n  8:  34921,  538416,  3560040, 13157856,  29665440,  41679360,  35551872,  16796160,   3359232;\n  9: 124373, 2149404, 16217424, 70004304, 190195776, 336685248, 387618048, 279376128, 114213888, 20155392;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+6*x)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 12*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 6*x)*P[n - 2, x] + (3 + 6*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; v[2] = 5 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 6*x)*v[i - 2] + (3 + 6*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-e646c30d067149e7","steps":["basis A013610"],"edges":["e-d7c15f9b4df77eea"],"text":"A249252 \u2192 basis A013610"},"headline":"A249252 \u2192 basis A013610","chain":null}