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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":"95703a5f14423a8de95f2b9b5e6f040484eaf5319db3171f3119c6e34c89432a","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-15df20466f2d1f05","role":"sequence","name":"Stirling-like number triangle defined by sequence A000217.","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:    4,       2;\n  3:   13,      20,        3;\n  4:   40,     146,       60,        4;\n  5:  121,     956,      819,      140,       5;\n  6:  364,    5978,     9624,     3192,     280,       6;\n  7: 1093,   36596,   105207,    60712,    9870,     504,     7;\n  8: 3280,  221762,  1106964,  1050956,  283160,   25956,   840,    8;\n  9: 9841, 1337132, 11402283, 17240292, 7260055, 1066560, 60522, 1320, 9;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = -(6+4*x+2*x^2)*P(n-2,x) - (6*x+5*x^2)*d/dx P(n-2,x) - (x^2+x^3)*d^2/dx^2 P(n-2,x) + (8+4*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) + x^2*d^2/dx^2 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[(-(6 + 4*x + 2*x^2)*P[n - 2, x] - (6*x + 5*x^2)*D[P[n - 2, x], x] - (x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (8 + 4*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}])/(2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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