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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":"36369441aa1198511df74ed95aed41c5507d5dd2ee3b4a27fd4f288309b4761c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8cf5a16a6dfee934","role":"sequence","name":"Triangle read by rows, k!*2^k*S_2(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":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:    1;\n  2:    4,      4;\n  3:   13,     36,       18;\n  4:   40,    232,      288,       96;\n  5:  121,   1320,     3060,     2400,       600;\n  6:  364,   7084,    27360,    37920,     21600,      4320;\n  7: 1093,  36876,   223398,   487200,    474600,    211680,    35280;\n  8: 3280, 188752,  1729728,  5576256,   8265600,   6168960,  2257920,   322560;\n  9: 9841, 956880, 12957480, 59411520, 125773200, 139708800, 84248640, 26127360, 3265920;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3+4*x+2*x^2)*P(n-2,x) - (2*x+3*x^2+x^3)*d/dx P(n-2,x) + (4+4*x)*P(n-1,x) + (2*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(3 + 4*x + 2*x^2)*P[n - 2, x] - (2*x + 3*x^2 + x^3)*D[P[n - 2, x], x] + (4 + 4*x)*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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