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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"717d91ed061447be10d28357d1043435a55838b75eaa8ef8a8652f4dd28f9425","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-656046abebe1865b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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:           4,             2;\n  2:          28,            38,             8;\n  3:         280,           776,           404,             48;\n  4:        3640,         18536,         17924,           5008,            384;\n  5:       58240,        518336,        830624,         409608,          71584,          3840;\n  6:     1106560,      16767296,      41936864,       32495928,        9711424,       1162368,         46080;\n  7:    24344320,     619114496,    2331585344,     2666855488,     1212434064,     242897792,      21157632,       645120;\n  8:   608608000,   25773203456,  142806196544,   231985816128,   150576319504,   44903213184,    6446438144,    426903552,   10321920;\n  9: 17041024000, 1196522302976, 9603514241024, 21593348460288, 19233376640704, 8008952906784, 1684973424896, 181779033600, 9459634176, 185794560;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+1+2*n*x)*P(n-1,x) + 2*n*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[(3*n + 1 + 2*n*x)*P[n - 1, x] + 2*n*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] = (3*n + 1 + 2*n*x)*v[i - 1] + 2*n*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-4d60c3dd8627c01d","steps":["basis A013609","basis A008277"],"edges":["e-23134a27b4e4bcc8","e-49fbb0ab2c08784a"],"text":"A094638 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A094638 \u2192 basis A013609 \u2192 basis A008277","chain":null}