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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"a0df73c86f69580ae048ab4fb21397062d04239799908f01f87feeae7934e93d","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-cf9f646a76b9c283","role":"sequence","name":"A triangular array related to A077028 and distributing the values of A007582.","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:      3,       2;\n  2:     10,      17,        4;\n  3:     36,     117,       65,        8;\n  4:    136,     755,      745,      209,       16;\n  5:    528,    4737,     7415,     3566,      609,       32;\n  6:   2080,   29267,    68509,    50365,    14420,     1665,      64;\n  7:   8256,  179097,   605255,   639758,   273525,    52123,    4353,    128;\n  8:  32896, 1089035,  5194165,  7602189,  4559436,  1276338,  173859,  11009,   256;\n  9: 131328, 6593697, 43705055, 86374646, 69897429, 26982186, 5333706, 545836, 27137, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8+10*x+2*x^2)*P(n-2,x) - (10*x+4*x^2)*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(8 + 10*x + 2*x^2)*P[n - 2, x] - (10*x + 4*x^2)*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x] + 3*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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 10*x + 2*x^2)*v[i - 2] - (10*x + 4*x^2)*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (6 + 3*x)*v[i - 1] + 3*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-cf9f646a76b9c283","steps":["basis A008277"],"edges":["e-b6b188019b41b535"],"text":"A110552 \u2192 basis A008277"},"headline":"A110552 \u2192 basis A008277","chain":null}