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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":"f1e70d0e99540a9a4684619e664b8c193849dfde9f88094c158b89bcce17c411","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-551f44a89cf7b3e1","role":"sequence","name":"basis_map_A013610 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:        3;\n  2:       36,        18;\n  3:      318,       324,        81;\n  4:     2484,      3834,      1944,       324;\n  5:    18153,     37584,     28836,      9720,      1215;\n  6:   127224,    330588,    340200,    173340,     43740,     4374;\n  7:   866364,   2709288,   3500010,   2391120,    911250,   183708,    15309;\n  8:  5777208,  21123324,  32851656,  28173420,  14390460,  4378374,   734832,   52488;\n  9: 37913199, 158693904, 288664884, 298056024, 191081835, 77892192, 19717992, 2834352, 177147;\n","recurrence_sentence":"The row polynomials satisfy (6*n-6)*P(n,x) = (18*n+36+(9*n+18)*x)*P(n-1,x) + (32+36*x+9*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n + 36 + (9*n + 18)*x)*P[n - 1, x] + (32 + 36*x + 9*x^2)*D[P[n - 1, x], x])/(6*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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