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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"97640f3294dd4128147351503329accc13ae1820bc826eb66f85abdfac488b48","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-08ac2c6d9914754c","role":"sequence","name":"Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), 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:       1;\n  1:       2,        1;\n  2:       6,        7,         1;\n  3:      24,       51,        17,         1;\n  4:     120,      417,       239,        34,        1;\n  5:     720,     3891,      3413,       829,       60,        1;\n  6:    5040,    41013,     51563,     18808,     2359,       97,      1;\n  7:   40320,   485163,    840725,    428221,    80380,     5810,    147,     1;\n  8:  362880,  6371505,  14807375,  10082350,  2624269,   285746,  12810,   212,   1;\n  9: 3628800, 92243907, 282129341, 249078973, 85596676, 12921771, 880634, 25878, 294, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-n+(n^2-n)*x)*P(n-2,x) + (n^2-n)*x*d/dx P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(n^2 - n + (n^2 - n)*x)*P[n - 2, x] + (n^2 - n)*x*D[P[n - 2, x], x] + (2 + x)*P[n - 1, x] + 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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (n^2 - n + (n^2 - n)*x)*v[i - 2] + (n^2 - n)*x*deriv(v[i - 2], x) + (2 + x)*v[i - 1] + 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-08ac2c6d9914754c","steps":["basis A008277"],"edges":["e-9042c8867d1aa2a0"],"text":"A154986 \u2192 basis A008277"},"headline":"A154986 \u2192 basis A008277","chain":null}