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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"bcd321a6f62a9f1373ba187261206bf1c410eee4b21a8e3f9dc029b2b42c1f4a","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-e5e648370a9e4459","role":"sequence","name":"Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+1)^n.","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:        3,         1;\n  2:       21,        19,          3;\n  3:      147,       253,         96,          9;\n  4:     1029,      2971,       2007,        432,         27;\n  5:     7203,     32797,      35004,      12933,       1809,        81;\n  6:    50421,    349579,     553443,     311940,      73170,      7209,      243;\n  7:   352947,   3647053,    8243496,    6651369,    2326050,    378108,    27702,     729;\n  8:  2470629,  37529371,  118106607,  131152392,   64149057,  15296526,  1826874,  103518,   2187;\n  9: 17294403, 382705597, 1647974004, 2452758093, 1612289259, 528970743, 91561428, 8379126, 378351, 6561;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(3*n-3)*x)*P(n-1,x) + (3*n-3)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (3*n - 3)*x)*P[n - 1, x] + (3*n - 3)*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (3*n - 3)*x)*v[i - 1] + (3*n - 3)*x*deriv(v[i - 1], x))/(n - 1)); 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-e5e648370a9e4459","steps":["basis A008277"],"edges":["e-a78f78fa25a65fe2"],"text":"A193728 \u2192 basis A008277"},"headline":"A193728 \u2192 basis A008277","chain":null}