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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"b66f3ec55dbbfcec5eb47c5513afde61ca250b40b2d7de8a0bcd62e30770e856","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-2b477f2aa77e3efc","role":"sequence","name":"Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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:         2,           1;\n  2:         8,          13,            3;\n  3:        48,         183,          113,            15;\n  4:       384,        3081,         3767,          1226,           105;\n  5:      3840,       61995,       133205,         79265,         15864,          945;\n  6:     46080,     1468125,      5210915,       5032180,       1760299,       237819,        10395;\n  7:    645120,    40238415,    227522225,     334447435,     181598074,     41671588,      4047987,      135135;\n  8:  10321920,  1257067665,  11069598575,   23814126910,   18818165149,   6516085618,   1054160442,    77071140,    2027025;\n  9: 185794560, 44172840915, 596989276205, 1831357932565, 2023202360284, 991065626187, 237272708546, 28479473694, 1622371230, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*d/dx 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 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*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-2b477f2aa77e3efc","steps":["basis A008277"],"edges":["e-450e8f8733e7ad5a"],"text":"A109692 \u2192 basis A008277"},"headline":"A109692 \u2192 basis A008277","chain":null}