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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"86a5a0401f264621dbf936f2e313bcbccdaf4475f7b1b27a7d4a7454c31b45a7","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-5fa21c8779d2bdd1","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:     11,      16,        3;\n  3:     38,      99,       40,        3;\n  4:    131,     550,      384,       70,       3;\n  5:    490,    3197,     3400,     1046,     106,       3;\n  6:   1897,   18698,    28711,    13278,    2277,     148,      3;\n  7:   7714,  113063,   240718,   155773,   39548,    4305,    196,     3;\n  8:  32501,  700540,  2019414,  1755990,  615028,   98168,   7394,   250,   3;\n  9: 142142, 4472521, 17130366, 19437464, 8981288, 1955984, 214090, 11844, 310, 3;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+(n+1)*x)*P(n-2,x) + (n+1)*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[(n + 1 + (n + 1)*x)*P[n - 2, x] + (n + 1)*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] = (n + 1 + (n + 1)*x)*v[i - 2] + (n + 1)*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-5fa21c8779d2bdd1","steps":["basis A008277"],"edges":["e-b777d60665436f83"],"text":"A249248 \u2192 basis A008277"},"headline":"A249248 \u2192 basis A008277","chain":null}