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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":"dd52a4704c34b2274e2bf2ffe45c4f2cc39041266effebd19e8a63ce739de072","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-f1291d4c6c67167e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.","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:  3,   2;\n  3:  5,  10,    3;\n  4:  7,  18,    6;\n  5: 11,  52,   57,   12;\n  6: 15,  86,  108,   24;\n  7: 23, 208,  489,  372,   60;\n  8: 31, 330,  870,  720,  120;\n  9: 47, 736, 3045, 4740, 2760, 360;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+5*x+3*x^2-2*x^3)*P(n-4,x) - (x+2*x^2-x^4)*d/dx P(n-4,x) + (3+5*x-x^2)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 3 + 2*x, P(3,x) = 5 + 10*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 3 + 2*x;\nP[3, x_] := 5 + 10*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(2 + 5*x + 3*x^2 - 2*x^3)*P[n - 4, x] - (x + 2*x^2 - x^4)*D[P[n - 4, x], x] + (3 + 5*x - x^2)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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