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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"f391a25121a37890382da51b95725fee000c132ddfc75da64a35875570f6bbdf","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-dd70177049267f47","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:      26,       17,         2;\n  1:      97,      109,        29,         2;\n  2:     362,      625,       281,        43,         2;\n  3:    1351,     3378,      2282,       580,        59,        2;\n  4:    5042,    17616,     16789,      6299,      1050,       77,        2;\n  5:   18817,    89744,    116068,     60302,     14640,     1741,       97,       2;\n  6:   70226,   449921,    769363,    531883,    176372,    30232,     2709,     119,      2;\n  7:  262087,  2230087,   4950031,   4432242,   1928219,   446719,    57225,    4016,    143,    2;\n  8:  978122, 10962601,  31160910,  35443842,  19681622,  5918458,  1016260,  101282,   5730,  169,   2;\n  9: 3650401, 53561040, 192978030, 274835562, 190968599, 72501025, 16023833, 2126346, 169899, 7925, 197, 2;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 26 + 17*x + 2*x^2, P(1,x) = 97 + 109*x + 29*x^2 + 2*x^3.","mathematica":"P[0, x_] := 26 + 17*x + 2*x^2;\nP[1, x_] := 97 + 109*x + 29*x^2 + 2*x^3;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 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] = 26 + 17*x + 2*x^2; v[2] = 97 + 109*x + 29*x^2 + 2*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 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-dfab35176bd08e72","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-23e971f1ed1ce0e7","e-a4c5480a42c33dcc"],"text":"A207606 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A207606 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}