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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":"4fb5c4425c6e926ae335f1ab8e45d6af134a611279bfdeeb865ca65f5afc40a8","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-8229dbdd6a4b5cd9","role":"sequence","name":"Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:         17,          9,          1;\n  3:        146,         96,         18,         1;\n  4:       1686,       1268,        309,        30,        1;\n  5:      24484,      20244,       5825,       755,       45,       1;\n  6:     428548,     380796,     123309,     19255,     1560,      63,      1;\n  7:    8780400,    8263576,    2927820,    522564,    51730,    2877,     84,    1;\n  8:  206138000,  203434248,   77439424,  15310736,  1752009,  120316,   4886,  108,   1;\n  9: 5455930400, 5605224048, 2263678596, 485725260, 62138979, 4975425, 251538, 7794, 135, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(6*n-8)*x+x^2)*P(n-2,x) + (6*n-5+2*x)*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[-(9*n^2 - 24*n + 16 + (6*n - 8)*x + x^2)*P[n - 2, x] + (6*n - 5 + 2*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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