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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":"a011d6ab079b214ed1abe3057c910705374e2083b4845a809896574aa62809ac","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-ca8f77e380f6014f","role":"sequence","name":"basis_map_A013610 of Riordan array (1-x,x/(1-x)).","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:    6,     18;\n  1:   15,     72,      81;\n  2:   36,    234,     486,     324;\n  3:   84,    684,    2025,    2592,     1215;\n  4:  192,   1872,    7128,   13284,    12150,     4374;\n  5:  432,   4896,   22680,   55080,    74115,    52488,    15309;\n  6:  960,  12384,   67392,  200880,   354780,   371790,   214326,    52488;\n  7: 2112,  30528,  190512,  671328,  1462860,  2020788,  1729917,   839808,   177147;\n  8: 4608,  73728,  518400, 2104704,  5443200,  9307872, 10532592,  7610760,  3188646,   590490;\n  9: 9984, 175104, 1368576, 6283008, 18779040, 38211264, 53642736, 51333264, 32063607, 11809800, 1948617;\n","recurrence_sentence":"The row polynomials satisfy (n+3+(3*n+3)*x)*P(n,x) = (2*n+8+(9*n+24)*x+(9*n+18)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 6 + 18*x.","mathematica":"P[0, x_] := 6 + 18*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (9*n + 24)*x + (9*n + 18)*x^2)*P[n - 1, x])/(n + 3 + (3*n + 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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