{"id":"n-7fd1278b59a5838a","revision":0,"first_index":0,"fingerprint":"f6-a7b8cffa591f9e51","name":"derivative of basis_map_A013609 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":"52c90bfe519e26c4d7afd9b5ad900138ed345d300ad0920235adfa73f516f01b","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-52d9fa43425cb9e1","role":"sequence","name":"basis_map_A013609 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:        0;\n  1:        4;\n  2:       20,        32;\n  3:      104,       288,       192;\n  4:      616,      2272,      2688,      1024;\n  5:     4176,     18560,     29760,     20480,      5120;\n  6:    32112,    163328,    319680,    302080,    138240,     24576;\n  7:   277056,   1563520,   3537408,   4121600,   2616320,    860160,    114688;\n  8:  2654784,  16288128,  41217792,  55960576,  44154880,  20299776,   5046272,   524288;\n  9: 27999360, 184119552, 509906688, 779433984, 721351680, 414941184, 145195008, 28311552, 2359296;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+8*n*x+16*x^2)*P(n-2,x) + (2*n+1+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 8*n*x + 16*x^2)*P[n - 2, x] + (2*n + 1 + 8*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] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 8*n*x + 16*x^2)*v[i - 2] + (2*n + 1 + 8*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 A013609","D"],"edges":["e-7070cc8d62757598","e-614882a4cbf0e295"],"text":"A137320 \u2192 basis A013609 \u2192 D"},"headline":"A137320 \u2192 basis A013609 \u2192 D","chain":null}