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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"09296c86b82e2db67b1f704d010ef8dd170c505346ac1725fe957ea82ef4c7c0","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-dfab35176bd08e72","role":"sequence","name":"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:    13,      13,        2;\n  1:    34,      63,       23,        2;\n  2:    89,     273,      176,       35,        2;\n  3:   233,    1118,     1130,      384,       49,        2;\n  4:   610,    4432,     6592,     3399,      725,       65,       2;\n  5:  1597,   17220,    36262,    26602,     8425,     1243,      83,       2;\n  6:  4181,   66045,   191938,   192475,    84852,    18304,    1988,     103,      2;\n  7: 10946,  251141,   989473,  1320186,   778014,   230041,   36113,    3016,    125,    2;\n  8: 28657,  949465,  5006568,  8718114,  6681432,  2600038,  553070,   66170,   4389,  149,   2;\n  9: 75025, 3575376, 24992832, 55995066, 54710124, 27251695, 7541555, 1211754, 114324, 6175, 175, 2;\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 >= 2, with P(0,x) = 13 + 13*x + 2*x^2, P(1,x) = 34 + 63*x + 23*x^2 + 2*x^3.","mathematica":"P[0, x_] := 13 + 13*x + 2*x^2;\nP[1, x_] := 34 + 63*x + 23*x^2 + 2*x^3;\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, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 13*x + 2*x^2; v[2] = 34 + 63*x + 23*x^2 + 2*x^3; for(n = 2, N, my(i = n + 1); 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-dfab35176bd08e72","steps":["basis A008277"],"edges":["e-d84114ea7bbb5d21"],"text":"A207606 \u2192 basis A008277"},"headline":"A207606 \u2192 basis A008277","chain":null}