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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-03f93edf37cf119a"]},{"id":198233,"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-03f93edf37cf119a"]},{"id":198234,"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 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2)","lean_route":null,"lean_declarations":[],"certificate_hash":"06e63c415c58d65136f701da60cb6f22526e994e8b4a64efb8cbbe64e997f279","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-a8292da3a4e72c99","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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:               1849,                  3776,                    1024;\n  1:              79507,                342368,                  230400,                   32768;\n  2:            3418801,              28221824,                35608576,                11927552,                 1048576;\n  3:          147008443,            2226038432,              4713216000,              2797404160,               560988160,                33554432;\n  4:         6321363049,          171657152576,            575547341824,            539662090240,            185445908480,             24763170816,              1073741824;\n  5:       271818611107,        13076570060768,          67076594457600,          93430545481728,          48980437237760,          10961192747008,           1044750794752,            34359738368;\n  6:     11688200277601,       989440950113024,        7595645848907776,       15133296844603392,       11365432223072256,        3792496119250944,         596274604670976,         42606075576320,         1099511627776;\n  7:    502592611936843,     74582093667360032,      844396216236748800,     2346588928564920320,     2427754409172664320,     1133570543346253824,      261484407913709568,      30456609528348672,      1692148395147264,       35184372088832;\n  8:  21611482313284249,   5609739988633981376,    92737022134687642624,   353196539990099886080,   490236849682603048960,   307802961392814784512,    97723093246801870848,   16499415797307801600,   1480563875056189440,    65794775806115840,    1125899906842624;\n  9: 929293739471222707, 421422066581573699168, 10102373048047865164800, 52061903766933888729088, 95132790575408317726720, 78171580352584698822656, 32814621677568512753664, 7532483001778842894336, 970669904155650293760, 69156114793622405120, 2514134491979579392, 36028797018963968;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (43+32*x)*P(n-1,x) + 32*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1849 + 3776*x + 1024*x^2.","mathematica":"P[0, x_] := 1849 + 3776*x + 1024*x^2;\nP[n_, x_] := P[n, x] = Expand[(43 + 32*x)*P[n - 1, x] + 32*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] = 1849 + 3776*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = (43 + 32*x)*v[i - 1] + 32*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-26633b980c3508e8","steps":["basis A013611","basis A008277"],"edges":["e-376f2b9433b6ff6b","e-03f93edf37cf119a"],"text":"A038226 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A038226 \u2192 basis A013611 \u2192 basis A008277","chain":null}