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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":"b12f65515999c19d79f41c0cb3e3cc6ee7c5a02096532ece3391c2a75f9a008b","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-f1c436fe49d0588b","role":"sequence","name":"basis_map_A008277 of Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","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:          1;\n  2:         12,          2;\n  3:        144,         48,          3;\n  4:       1920,        960,        120,         4;\n  5:      28800,      19200,       3600,       240,        5;\n  6:     483840,     403200,     100800,     10080,      420,       6;\n  7:    9031680,    9031680,    2822400,    376320,    23520,     672,     7;\n  8:  185794560,  216760320,   81285120,  13547520,  1128960,   48384,  1008,    8;\n  9: 4180377600, 5573836800, 2438553600, 487710720, 50803200, 2903040, 90720, 1440, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n^2+2*n+n*x)*P(n-1,x) + 2*n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 2*n + n*x)*P[n - 1, x] + 2*n*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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