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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":"4b50b4d11dacebcebed2eee1a806b8f8420cac21db96bd0aef3c72833a615d69","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-ff0214c333bfdcca","role":"sequence","name":"Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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:           1;\n  1:          16,           1;\n  2:         255,          32,          1;\n  3:        4050,         765,         48,          1;\n  4:       64125,       16200,       1530,         64,         1;\n  5:     1012500,      320625,      40500,       2550,        80,       1;\n  6:    15946875,     6075000,     961875,      81000,      3825,      96,      1;\n  7:   250593750,   111628125,   21262500,    2244375,    141750,    5355,    112,    1;\n  8:  3929765625,  2004750000,  446512500,   56700000,   4488750,  226800,   7140,  128,   1;\n  9: 61509375000, 35367890625, 9021375000, 1339537500, 127575000, 8079750, 340200, 9180, 144, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+14+x)*P(n,x) = (15*n+225+(n+30)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n + 225 + (n + 30)*x + x^2)*P[n - 1, x])/(n + 14 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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