{"id":"n-c23a55e8e9133b7b","revision":0,"first_index":0,"fingerprint":"f6-822ca06cf4d0422c","name":"one_plus_derivative of basis_map_A013610 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. 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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"065972c4ae3b92f39865ad1ca0aa808df8d85b6672efa0fdf2d97a186e612113","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8a9c0145328414df","role":"sequence","name":"basis_map_A013610 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= 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:        8,         6;\n  2:       36,       102,         36;\n  3:      180,       804,        972,        216;\n  4:     1044,      6036,      11628,       8208,       1296;\n  5:     6984,     48024,     121320,     137160,      64800,       7776;\n  6:    53208,    415656,    1262664,    1888920,    1432080,     489888,      46656;\n  7:   455904,   3933504,   13697712,   24845184,   25084080,   13771296,    3592512,     279936;\n  8:  4345056,  40630464,  157434192,  329670432,  406126224,  298287360,  124758144,   25754112,   1679616;\n  9: 45627840, 456268032, 1928069568, 4519529568, 6464235456, 5821992288, 3268532736, 1079993088, 181398528, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+12*n*x+36*x^2)*P(n-2,x) + (2*n+1+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 12*n*x + 36*x^2)*P[n - 2, x] + (2*n + 1 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 12*n*x + 36*x^2)*v[i - 2] + (2*n + 1 + 12*x)*v[i - 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-60c7e36cbe887b7e","steps":["basis A013610","D+1"],"edges":["e-881cf76aa24e5971","e-6f654a6a60bb5fcb"],"text":"A137320 \u2192 basis A013610 \u2192 D+1"},"headline":"A137320 \u2192 basis A013610 \u2192 D+1","chain":null}