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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"13837fc85d8397908a733fb06338a060697f1589b7ef9a5c3ab54a41a2af4f9c","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-2a67b0774691ba3b","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.","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:           2;\n  1:          26,           13;\n  2:         338,          340,             2;\n  3:        4394,         8794,          2207,            2;\n  4:       57122,       176350,         67100,         2503,            2;\n  5:      742586,      3729598,       2649095,       396360,         2801,           2;\n  6:     9653618,     70535926,      68329928,     15657911,       772525,        3101,         2;\n  7:   125497034,   1394116222,    2013649907,    752157988,     78975256,     1196491,      3403,       2;\n  8:  1631461442,  25861587910,   48848892380,  24501944643,   4017446172,   200482426,   1669160,    3707,    2;\n  9: 21208998746, 495879540526, 1284471587279, 889544921432, 212632521999, 17740874040, 388340322, 2191440, 4013, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (143*n-286+(143*n-286)*x)*P(n-2,x) + (143*n-286)*x*d/dx P(n-2,x) + (2*n-4+(n-2)*x)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 26 + 13*x, P(2,x) = 338 + 340*x + 2*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 26 + 13*x;\nP[2, x_] := 338 + 340*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((143*n - 286 + (143*n - 286)*x)*P[n - 2, x] + (143*n - 286)*x*D[P[n - 2, x], x] + (2*n - 4 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 26 + 13*x; v[3] = 338 + 340*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((143*n - 286 + (143*n - 286)*x)*v[i - 2] + (143*n - 286)*x*deriv(v[i - 2], x) + (2*n - 4 + (n - 2)*x)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 2)); 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-2a67b0774691ba3b","steps":["basis A008277"],"edges":["e-732d1ecf5e63d399"],"text":"A153651 \u2192 basis A008277"},"headline":"A153651 \u2192 basis A008277","chain":null}