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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"5f7c881f8ca5477ade767341800edc84ac0d8360d0ea3ad6924e9041869c94bc","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-1bffaa6198344f93","role":"sequence","name":"Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:      4,        2;\n  2:     18,       22,         4;\n  3:     80,      184,        80,         8;\n  4:    356,     1388,      1076,       248,        16;\n  5:   1584,     9936,     12232,      4856,       704,       32;\n  6:   7048,    69048,    126960,     77160,     18816,     1888,      64;\n  7:  31360,   471296,   1247040,   1088832,    396864,    65856,    4864,    128;\n  8: 139536,  3180784,  11818768,  14253600,   7282512,  1776000,  214208,  12160,   256;\n  9: 620864, 21311680, 109336544, 177378272, 122043648, 41013024, 7179136, 659072, 29696, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + 2*x*d/dx P(n-2,x) + (4+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + (4 + 2*x)*P[n - 1, x] + 2*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)); 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