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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":"ac08520567d127d20e263007fdc04fa532c347b2f48ce52125c67e829a2e7d66","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-02f1a3354aa99430","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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:      0;\n   2:      4;\n   3:     24,      32;\n   4:    100,     288,     192;\n   5:    360,    1632,    2304,     1024;\n   6:   1200,    7488,   16512,    15360,     5120;\n   7:   3816,   30432,   92160,   133120,    92160,    24576;\n   8:  11756,  114336,  442560,   875520,   936960,   516096,   114688;\n   9:  35400,  406624, 1921536,  4853760,  7096320,  6021120,  2752512,   524288;\n  10: 104780, 1388736, 7761792, 23933952, 44620800, 51609600, 36241408, 14155776, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (8*n-16)*P(n,x) = (12*n+12+(16*n+16)*x)*P(n-1,x) + (5+24*x+16*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n + 12 + (16*n + 16)*x)*P[n - 1, x] + (5 + 24*x + 16*x^2)*D[P[n - 1, x], x])/(8*n - 16)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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