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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":"d50f79fdf2197fd6288e4906c0ae4aa25bb50eda4195044af645097ad7cc773a","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-a1e0668357b9b581","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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:        4;\n  2:       48,        32;\n  3:      424,       576,       192;\n  4:     3312,      6816,      4608,      1024;\n  5:    24204,     66816,     68352,     30720,      5120;\n  6:   169632,    587712,    806400,    547840,    184320,     24576;\n  7:  1155152,   4816512,   8296320,   7557120,   3840000,   1032192,    114688;\n  8:  7702944,  37552576,  77870592,  89041920,  60641280,  24600576,   5505024,   524288;\n  9: 50550932, 282122496, 684242688, 942004224, 805217280, 437649408, 147718144, 28311552, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (3*n+6+(2*n+4)*x)*P(n-1,x) + (4+6*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 6 + (2*n + 4)*x)*P[n - 1, x] + (4 + 6*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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