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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":"011ab7188e82809cdfc7ff5d61797e67327314844890ba89e1e1b15ca68f8d72","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-170093e6e241015d","role":"sequence","name":"Triangle read by rows, 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:       1;\n   2:       5,         3;\n   3:      25,        57,         18;\n   4:     125,       657,        756,         162;\n   5:     625,      6333,      15498,       11988,         1944;\n   6:    3125,     56193,     242820,      383130,       223560,        29160;\n   7:   15625,    477477,    3326778,     8846820,     10264320,      4811400,       524880;\n   8:   78125,   3960057,   42230916,   172371402,    327544560,    300056400,    117573120,     11022480;\n   9:  390625,  32382813,  511592778,  3035017188,   8590362984,  12641151600,   9560164320,   3218564160,   264539520;\n  10: 1953125, 262576593, 6013772100, 50068154970, 199675287000, 428915927160, 513823927680, 330740534880, 97615082880, 7142567040;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(10+18*x+3*x^2-9*x^3)*P(n-2,x) - (6*x+6*x^2-9*x^3-9*x^4)*d/dx P(n-2,x) + (7+9*x-3*x^2)*P(n-1,x) + (3*x+3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(10 + 18*x + 3*x^2 - 9*x^3)*P[n - 2, x] - (6*x + 6*x^2 - 9*x^3 - 9*x^4)*D[P[n - 2, x], x] + (7 + 9*x - 3*x^2)*P[n - 1, x] + (3*x + 3*x^2)*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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