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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"cde29c900ebeada05ae4d1224f415e08a0db40aa3e16ce926f0613798f6aa6a9","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-9e472291b944d56e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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:            23,             12;\n  2:           529,            696,             144;\n  3:         12167,          30708,           15120,            1728;\n  4:        279841,        1220784,         1079136,          283392,           20736;\n  5:       6436343,       46085532,        65368800,        29669760,         4872960,          248832;\n  6:     148035889,     1690229736,      3625359984,      2534941440,       702017280,        79128576,        2985984;\n  7:    3404825447,    60934471428,    190674676080,    193065864768,     80262524160,     14991879168,     1233211392,      35831808;\n  8:   78310985281,  2173564405344,   9692923432896,  13678982134272,   8015429592576,   2207476260864,   297057632256,   18632540160,    429981696;\n  9: 1801152661463, 77014486010412, 481650174210240, 923375027116800, 733243286684160, 279405684762624, 54710190194688, 5558373384192, 274758303744, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+12*x)*P(n-1,x) + 12*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[(23 + 12*x)*P[n - 1, x] + 12*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] = (23 + 12*x)*v[i - 1] + 12*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-9e472291b944d56e","steps":["basis A008277"],"edges":["e-aadd11f319a04cdb"],"text":"A038326 \u2192 basis A008277"},"headline":"A038326 \u2192 basis A008277","chain":null}