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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9d401c5737a19de9c56201d3ca673490fb591c1f75eeab4b466f3309c2ed71a9","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-016d7724d51f6416","role":"sequence","name":"Riordan array (1, 3+3x).","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,        3;\n  2:     12,       30,         9;\n  3:     45,      243,       180,         27;\n  4:    171,     1809,      2520,        891,         81;\n  5:    648,    12960,     30456,      19116,       3969,       243;\n  6:   2457,    91071,    341091,     339012,     120771,     16524,       729;\n  7:   9315,   633501,   3656016,    5412177,    2945484,    676026,     65610,     2187;\n  8:  35316,  4382748,  38147679,   81005589,   63247392,  21664665,   3470769,   251505,   6561;\n  9: 133893, 30231387, 391282002, 1162424277, 1252146438, 600711066, 141286032, 16697745, 938223, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+3*x)*P(n-2,x) + 3*x*d/dx P(n-2,x) + (3+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 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 2, x] + 3*x*D[P[n - 2, x], x] + (3 + 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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 2] + 3*x*deriv(v[i - 2], x) + (3 + 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-016d7724d51f6416","steps":["basis A008277"],"edges":["e-84ef3497fd0acbc4"],"text":"A099093 \u2192 basis A008277"},"headline":"A099093 \u2192 basis A008277","chain":null}