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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":"46a37500c3faacf957b6ac270c492e28a0f55c4c8274800edd2599f2d654cd2e","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-67a3afb389f8b05c","role":"sequence","name":"basis_map_A013609 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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:         32,          32;\n   4:        284,         480,         192;\n   5:       2976,        6592,        4608,        1024;\n   6:      36516,       96320,       90240,       35840,       5120;\n   7:     516288,     1544288,     1751040,      947200,     245760,      24576;\n   8:    8284860,    27294624,    35531328,    23546880,    8422400,    1548288,    114688;\n   9:  148950720,   530287488,   767800320,   589797376,  260915200,   66748416,   9175040,   524288;\n  10: 2967901380, 11267068032, 17769931008, 15301085184, 7906375680, 2520612864, 485818368, 51904512, 2359296;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(16*n-24)*x+16*x^2)*P(n-2,x) + (4*n-4+8*x)*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[-(4*n^2 - 12*n + 9 + (16*n - 24)*x + 16*x^2)*P[n - 2, x] + (4*n - 4 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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