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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":"ee5e60e96ecd744f1036c738a110266a7eec7cc7b821756b4221e4943e4ef33c","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-dabecbb1addaf41b","role":"sequence","name":"basis_map_A013610 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 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:       4,        3;\n  2:      11,       27,        9;\n  3:      39,      141,      135,        27;\n  4:     174,      780,     1125,       594,       81;\n  5:     942,     4872,     8910,      7155,     2430,      243;\n  6:    6012,    34524,    74151,     76545,    39690,     9477,      729;\n  7:   44244,   275580,   666477,    817803,   549990,   200718,    35721,    2187;\n  8:  369072,  2454984,  6513120,   9090144,  7329609,  3490452,   949158,  131220,   6561;\n  9: 3442608, 24188328, 69172380, 106800660, 98692020, 56709639, 20207880, 4264650, 472392, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(6*n-6)*x+9*x^2)*P(n-2,x) + (2*n-1+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (6*n - 6)*x + 9*x^2)*P[n - 2, x] + (2*n - 1 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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