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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":"57019eebb2614195c1fda48fccf513522be9b5d203282fa355e5d811efb39f03","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-cfa0bcf0bbd4ed6d","role":"sequence","name":"Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) terms per row.","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:   0;\n  1:   0;\n  2:   1;\n  3:   3;\n  4:   7,    2;\n  5:  15,   12;\n  6:  31,   50,    3;\n  7:  63,  180,   30;\n  8: 127,  602,  195,  4;\n  9: 255, 1932, 1050, 60;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 2*x*P(n-3,x) - x^2*d/dx P(n-3,x) - (2-2*x)*P(n-2,x) - x*d/dx P(n-2,x) + 3*P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 2*x*P[n - 3, x] - x^2*D[P[n - 3, x], x] - (2 - 2*x)*P[n - 2, x] - x*D[P[n - 2, x], x] + 3*P[n - 1, x] + 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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