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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"e7c4b008c87f6da556b6752f38c592d2eec2b23350ee1576f8a832df7666d207","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-27c21e0d08511c1c","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, 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:      10,        5;\n  2:      50,       52,         2;\n  3:     250,      506,       135,         2;\n  4:    1250,     4078,      1916,       175,        2;\n  5:    6250,    32414,     25407,      4936,      217,        2;\n  6:   31250,   244582,    281432,     89327,     9013,      261,      2;\n  7:  156250,  1831166,   2999835,   1405332,   233720,    14275,    307,     2;\n  8:  781250, 13455958,  30163676,  19607595,  4823532,   492330,  20856,   355,   2;\n  9: 3906250, 98427854, 296570727, 257519096, 87157767, 13060392, 905538, 28896, 405, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (15*n-30+(15*n-30)*x)*P(n-2,x) + (15*n-30)*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) = 10 + 5*x, P(2,x) = 50 + 52*x + 2*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 10 + 5*x;\nP[2, x_] := 50 + 52*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n - 30 + (15*n - 30)*x)*P[n - 2, x] + (15*n - 30)*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] = 10 + 5*x; v[3] = 50 + 52*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((15*n - 30 + (15*n - 30)*x)*v[i - 2] + (15*n - 30)*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-27c21e0d08511c1c","steps":["basis A008277"],"edges":["e-d127f77ad15cf466"],"text":"A153648 \u2192 basis A008277"},"headline":"A153648 \u2192 basis A008277","chain":null}