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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"ac90eecaca1e187ea1c451501e2db24bdb80081de3c9b9fb9da60237a3051bf3","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-d75669a8cd7f9393","role":"sequence","name":"A133080 * A007318.","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,     1;\n  2:   4,     5,      1;\n  3:  12,    24,     10,      1;\n  4:  16,    65,     55,     14,      1;\n  5:  48,   276,    340,    139,     21,     1;\n  6:  64,   665,   1351,    910,    245,    27,     1;\n  7: 192,  2724,   7420,   6811,   2625,   461,    36,    1;\n  8: 256,  6305,  26335,  35574,  20181,  5418,   714,   44,  1;\n  9: 768, 25476, 137980, 239779, 176841, 63525, 11844, 1154, 55, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+5*x+x^2)*P(n-2,x) + (5*x+2*x^2)*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(4 + 5*x + x^2)*P[n - 2, x] + (5*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 5*x + x^2)*v[i - 2] + (5*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), 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-d75669a8cd7f9393","steps":["basis A008277"],"edges":["e-3d4e8ad6d5c37e78"],"text":"A133087 \u2192 basis A008277"},"headline":"A133087 \u2192 basis A008277","chain":null}