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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"ee4c63482bc511b8ae136f6438e72df579ebf6ca59235d4cac5a56fbfb65963b","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-efce1b6c07b70e1a","role":"sequence","name":"Triangle of coefficients in expansion of (1+6x)^n.","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:        7,           6;\n  2:       49,         120,           36;\n  3:      343,        1854,         1404,          216;\n  4:     2401,       26160,        37800,        13824,         1296;\n  5:    16807,      354486,       875160,       572400,       123120,         7776;\n  6:   117649,     4709160,     18754956,     19560960,      7251120,      1026432,       46656;\n  7:   823543,    61924974,    384599124,    601553736,    342150480,     81484704,     8164800,     279936;\n  8:  5764801,   809965920,   7678933200,  17346438144,  14215987296,   5067836928,   839994624,   62705664,   1679616;\n  9: 40353607, 10564145766, 150759526320, 479734552800, 544774235040, 272805890112, 66526790400, 8112545280, 468612864, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+6*x)*P(n-1,x) + 6*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[(7 + 6*x)*P[n - 1, x] + 6*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] = (7 + 6*x)*v[i - 1] + 6*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-efce1b6c07b70e1a","steps":["basis A008277"],"edges":["e-7bcf400b3d75a9d5"],"text":"A013613 \u2192 basis A008277"},"headline":"A013613 \u2192 basis A008277","chain":null}