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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":"e75b039d8f8b6e516b52e0c16bb1a5eddae5fff1450c171162dd0f10f96b74ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8cf5a16a6dfee934","role":"sequence","name":"Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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:    2,      1;\n  2:    5,      8,        2;\n  3:   14,     49,       36,        6;\n  4:   41,    272,      404,      192,        24;\n  5:  122,   1441,     3720,     3420,      1200,       120;\n  6:  365,   7448,    30902,    47040,     31080,      8640,       720;\n  7: 1094,  37969,   241836,   561666,    596400,    306600,     70560,     5040;\n  8: 3281, 192032,  1824104,  6152832,   9659664,   7822080,   3286080,   645120,   40320;\n  9: 9842, 966721, 13435920, 63730680, 140626080, 164863440, 107533440, 38162880, 6531840, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3+2*x-x^2-x^3)*P(n-2,x) - (2*x-x^2-3*x^3-x^4)*d/dx P(n-2,x) + (4+2*x-x^2)*P(n-1,x) + (2*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(3 + 2*x - x^2 - x^3)*P[n - 2, x] - (2*x - x^2 - 3*x^3 - x^4)*D[P[n - 2, x], x] + (4 + 2*x - x^2)*P[n - 1, x] + (2*x + x^2)*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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