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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"05d1bdface5e748160f20ce28dcbb24ca06ba9383e10f39293f2d5e34afda4df","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-30cb3cad216ed189","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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:       5,       1;\n  2:      24,      11,       1;\n  3:     115,      89,      18,       1;\n  4:     551,     638,     214,      26,       1;\n  5:    2640,    4290,    2118,     421,      35,      1;\n  6:   12649,   27742,   18902,    5460,     735,     45,     1;\n  7:   60605,  174811,  157938,   62161,   12040,   1184,    56,    1;\n  8:  290376, 1081729, 1261475,  649766,  169786,  23835,  1799,   68,  1;\n  9: 1391275, 6605939, 9754116, 6397442, 2165800, 406952, 43568, 2614, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (5 + 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] = 5 + x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (5 + 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-30cb3cad216ed189","steps":["basis A008277"],"edges":["e-ae643fa072a3e124"],"text":"A207823 \u2192 basis A008277"},"headline":"A207823 \u2192 basis A008277","chain":null}