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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"9853d0bf69ff4ade6b8c50e8f9e17e9e63678a501b19f1208a45eb63009891a6","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-b8883330fe0b24f8","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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,           4;\n  2:         35,          58,           24;\n  3:        315,         802,          680,          192;\n  4:       3465,       11972,        15500,         8912,         1920;\n  5:      45045,      197216,       345164,       301856,       131904,        23040;\n  6:     675675,     3588870,      7938484,      9360136,      6204544,      2192256,       322560;\n  7:   11486475,    71821590,    192376148,    286138056,    255239424,    136541056,     40559616,     5160960;\n  8:  218243025,  1571366760,   4947935432,   8899393728,  10000034064,   7188589696,   3228371712,   828131328,    92897280;\n  9: 4583103525, 37363562460, 135333979272, 285845976928, 387988589904, 350961064896, 211567599872, 81958192128, 18513469440, 1857945600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+3+(2*n+2)*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[(2*n + 3 + (2*n + 2)*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] = (2*n + 3 + (2*n + 2)*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-2feb6dc01f424b0a","steps":["basis A007318","basis A013609"],"edges":["e-5281dccfa5f22cf2","e-6cae61e341fbf12c"],"text":"A145324 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A145324 \u2192 basis A007318 \u2192 basis A013609","chain":null}