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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":"4d4aeecbec3187ed32dce04c13257a46d74a204083c3027254525d9f1f190428","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:    1;\n  1:    2,       1;\n  2:    5,       6,        1;\n  3:   14,      33,       13,        1;\n  4:   41,     186,      133,       24,        1;\n  5:  122,    1077,     1297,      413,       40,       1;\n  6:  365,    6342,    12613,     6400,     1078,      62,      1;\n  7: 1094,   37689,   123505,    95781,    25048,    2478,     91,    1;\n  8: 3281,  225042,  1217845,  1419944,   545899,   82588,   5166,  128,   1;\n  9: 9842, 1346973, 12070849, 21041053, 11570128, 2518571, 238282, 9966, 174, 1;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = -(12-4*x-4*x^2+2*x^3)*P(n-2,x) - (12*x-10*x^2+4*x^3)*d/dx P(n-2,x) - (2*x^2-2*x^3+x^4)*d^2/dx^2 P(n-2,x) + (16-4*x+2*x^2)*P(n-1,x) + 12*x*d/dx P(n-1,x) + 2*x^2*d^2/dx^2 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[Together[(-(12 - 4*x - 4*x^2 + 2*x^3)*P[n - 2, x] - (12*x - 10*x^2 + 4*x^3)*D[P[n - 2, x], x] - (2*x^2 - 2*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (16 - 4*x + 2*x^2)*P[n - 1, x] + 12*x*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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