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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"b6e41240ae543d29e52033aec8e972b69017117a0f89b15fb425c56653e2e043","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-b0d4d18dfd70fd8a","role":"sequence","name":"basis_map_A013610 of Even row Pascal-square read by antidiagonals.","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:        9,          6;\n  2:       65,        111,          27;\n  3:      441,       1464,         846,         108;\n  4:     2929,      17022,       17604,        5130,         405;\n  5:    19305,     186615,      308583,      154332,       27135,        1458;\n  6:   126881,    1984320,     4945050,     3783618,     1113021,      130977,       5103;\n  7:   833049,   20744886,    75202560,    82774116,    36280710,     7021728,     591948,     17496;\n  8:  5467345,  214752831,  1106875827,  1687617720,  1035294534,   293329188,   40170816,   2545668,    59049;\n  9: 35877321, 2210106504, 15947166726, 32842427184, 27124393050, 10562320134, 2090960082, 213258744, 10530405, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16+33*x+9*x^2)*P(n-2,x) - (33*x+18*x^2)*d/dx P(n-2,x) - 9*x^2*d^2/dx^2 P(n-2,x) + (9+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(16 + 33*x + 9*x^2)*P[n - 2, x] - (33*x + 18*x^2)*D[P[n - 2, x], x] - 9*x^2*D[P[n - 2, x], {x, 2}] + (9 + 6*x)*P[n - 1, x] + 6*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] = 9 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(16 + 33*x + 9*x^2)*v[i - 2] - (33*x + 18*x^2)*deriv(v[i - 2], x) - 9*x^2*deriv(deriv(v[i - 2], x), x) + (9 + 6*x)*v[i - 1] + 6*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-79ebafb266da3b41","steps":["basis A013610","basis A008277"],"edges":["e-01dc3c5cce6c6f01","e-60e0628de39fb70b"],"text":"A172431 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A172431 \u2192 basis A013610 \u2192 basis A008277","chain":null}