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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":"952d861a10e38f25fcf7174e5f84737615f915b035f02aae7287427f5f499e21","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-181be2f2eb86cd8e","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 3).","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:       4,        1;\n  2:      18,       11,        1;\n  3:      90,       99,       21,        1;\n  4:     486,      855,      315,       34,        1;\n  5:    2754,     7371,     4185,      765,       50,       1;\n  6:   16038,    64071,    52731,    14490,     1575,      69,      1;\n  7:   94770,   562059,   648081,   251181,    40320,    2898,     91,    1;\n  8:  564246,  4968135,  7869555,  4140234,   920241,   96390,   4914,  116,   1;\n  9: 3372354, 44162091, 94992345, 66196845, 19578510, 2801547, 206010, 7830, 144, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(18-3*x-2*x^2)*P(n-2,x) - (9*x-6*x^2)*d/dx P(n-2,x) + (9-x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-(18 - 3*x - 2*x^2)*P[n - 2, x] - (9*x - 6*x^2)*D[P[n - 2, x], x] + (9 - x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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