{"id":"n-428b25ab66142992","revision":0,"first_index":0,"fingerprint":"f6-ef12d04f3c0ae80e","name":"basis_map_A008277 of basis_map_A007318 of Triangle whose (i,j)-th entry is 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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-b3c369449bb73430"]},{"id":198104,"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-b3c369449bb73430"]},{"id":198105,"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-b3c369449bb73430"]}],"region_exact":{},"incoming_edges":[{"id":"e-b3c369449bb73430","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-428b25ab66142992","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"d26768012cc1e721581cba56a7f82bb45d87aaa5d58b90975a13c2b0d20455f8","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-6ffee7a877aa82b8","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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":[],"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:             529,               312,                36;\n  1:           12167,             12222,              3132,               216;\n  2:          279841,            427440,            182952,             27648,              1296;\n  3:         6436343,          14074806,           8967960,           2231280,            226800,              7776;\n  4:       148035889,         446787432,         398327436,         145290240,          24047280,           1772928,            46656;\n  5:      3404825447,       13845050862,       16622184852,        8346864456,        2001963600,         238248864,         13390272,          279936;\n  6:     78310985281,      421935427680,      664846774992,      441954551808,      144173475936,       24638971392,       2219519232,        98537472,        1679616;\n  7:   1801152661463,    12705993314406,    25801249690800,    22109217274080,     9427880679840,     2170906339392,     278785463040,     19722051072,      710477568,      10077696;\n  8:  41426511213649,   379280722086552,   979279699064436,  1061285406382080,   575765695596960,   171625320066816,   29473780355712,   2954646097920,   168776213760,    5038848000,    60466176;\n  9: 952809757913927, 11247700007791902, 36550473799774572, 49388379856051896, 33428700131349600, 12550736137123008, 2768704961387904, 368894678499072, 29710987764480, 1400648578560, 35251780608, 362797056;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 529 + 312*x + 36*x^2.","mathematica":"P[0, x_] := 529 + 312*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(23 + 6*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 529 + 312*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 6*x)*v[i - 1] + 6*x*deriv(v[i - 1], x)); 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-1f2dec04776afb2b","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-653ba39db5056185","e-b3c369449bb73430"],"text":"A038320 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A038320 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}