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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"7991e07d48a2ca621802c757077e5360438761516f3a5f0c5ab52250a8670df0","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-14dfd91463cae288","role":"sequence","name":"Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","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,      3;\n  2:    9,     14,      3;\n  3:   22,     61,     29,       3;\n  4:   53,    242,    200,      47,      3;\n  5:  128,    923,   1190,     476,     68,      3;\n  6:  309,   3434,   6525,    3958,    946,     92,     3;\n  7:  746,  12585,  34027,   29409,  10450,   1676,   119,    3;\n  8: 1801,  45678, 171702,  203429, 100797,  23680,  2741,  149,   3;\n  9: 4348, 164751, 847152, 1340510, 889870, 287063, 48117, 4225, 182, 3;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + 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) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + 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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + 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-14dfd91463cae288","steps":["basis A008277"],"edges":["e-ac324c30ef49a129"],"text":"A102662 \u2192 basis A008277"},"headline":"A102662 \u2192 basis A008277","chain":null}