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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":"8897d568f1a85b70989fc31e1f5b9d6e063b1bb6f3eb42fac47855d039cd6a52","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-bdd5560e7eb65b04","role":"sequence","name":"Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       0;\n   2:       3;\n   3:      21,        18;\n   4:     117,       270,        81;\n   5:     609,      2862,      2106,        324;\n   6:    3093,     26550,     36045,      12960,       1215;\n   7:   15561,    230958,    515970,     325620,      69255,      4374;\n   8:   77997,   1941030,   6714981,    6622560,    2398410,    336798,    15309;\n   9:  390369,  15996222,  82599426,  119575764,   65607570,  15370236,  1530900,   52488;\n  10: 1952613, 130311990, 980576685, 2004458400, 1563737805, 543591972, 89006526, 6613488, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(10+12*x+9*x^2)*P(n-2,x) - (6*x+9*x^2)*d/dx P(n-2,x) + (7+6*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-(10 + 12*x + 9*x^2)*P[n - 2, x] - (6*x + 9*x^2)*D[P[n - 2, x], x] + (7 + 6*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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