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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"841971828cf8bbf837f4fe9f4141f32f7e7b8230b0693e282ca99516d9c47059","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-77900e9176a9461a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^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:              1;\n  1:             29,              33;\n  2:            841,            1914,            1089;\n  3:          24389,           83259,           94743,            35937;\n  4:         707281,         3219348,         5495094,          4168692,          1185921;\n  5:       20511149,       116701365,       265596210,        302230170,        171958545,         39135393;\n  6:      594823321,      4061207502,     11553435135,      17529349860,      14960393415,       6809558382,       1291467969;\n  7:    17249876309,    137404187151,    469069466481,     889614505395,    1012319954415,     691170175773,     262167997707,      42618442977;\n  8:   500246412961,   4553967345576,  18137352703932,   41278113050328,   58714557356070,   53450493593112,   30411487734012,    9887478770664,   1406408618241;\n  9: 14507145975869, 148573184649417, 676264150818036, 1795597917689268, 3064899893986854, 3487644706950558, 2645799432859044, 1290315979571652, 367072649360901, 46411484401953;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (29+33*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[(29 + 33*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] = (29 + 33*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-b7cf68162d40a956","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-9e4137eebf66e3b4","e-4b9feda15f3ab72a"],"text":"A038277 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038277 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}