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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"6d1ab4178a923283d278640c6282187a27bee14236b850462cfe90c6c4d7293b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-318700a428d7b97b","role":"sequence","name":"Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    4,      1;\n   3:   11,      8,      1;\n   4:   29,     42,     13,      1;\n   5:   76,    189,    106,     19,      1;\n   6:  199,    790,    706,    219,     26,      1;\n   7:  521,   3170,   4214,   2001,    400,     34,     1;\n   8: 1364,  12411,  23534,  16001,   4775,    671,    43,    1;\n   9: 3571,  47838, 125867, 117539,  49026,  10109,  1057,   53,  1;\n  10: 9349, 182512, 653639, 815100, 455946, 129227, 19579, 1586, 64, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (3 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A008277"],"edges":["e-38992239a80c1f97"],"text":"A111125 \u2192 basis A008277"},"headline":"A111125 \u2192 basis A008277","chain":null}