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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":"9ffb581b13d138cb92ca77420155bdb32ad72484930ef7e1b3f94322f416b88f","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-be3f02dc425ece52","role":"sequence","name":"basis_map_A007318 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:          2;\n   3:         16,          8;\n   4:        142,        120,         24;\n   5:       1488,       1648,        576,        64;\n   6:      18258,      24080,      11280,      2240,       160;\n   7:     258144,     386072,     218880,     59200,      7680,      384;\n   8:    4142430,    6823656,    4441416,   1471680,    263200,    24192,     896;\n   9:   74475360,  132571872,   95975040,  36862336,   8153600,  1042944,   71680,   2048;\n  10: 1483950690, 2816767008, 2221241376, 956317824, 247074240, 39384576, 3795456, 202752, 4608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(8*n-12)*x+4*x^2)*P(n-2,x) + (4*n-4+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (8*n - 12)*x + 4*x^2)*P[n - 2, x] + (4*n - 4 + 4*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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