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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":"2e99c145739935aefb201ef706a39bd19365993e0ddd278170fd8164a82aea53","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-048e77154c1b7b9a","role":"sequence","name":"shift_x_plus_one of A Jacobsthal triangle.","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:   11,    14,     3;\n  4:   30,    56,    30,     4;\n  5:   73,   188,   162,    52,     5;\n  6:  176,   566,   672,   356,    80,    6;\n  7:  407,  1586,  2409,  1788,   665,  114,    7;\n  8:  930,  4236,  7830,  7504,  3950, 1116,  154,   8;\n  9: 2085, 10896, 23760, 27944, 19130, 7680, 1736, 200, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n+1-(n-1)*x)*P(n,x) = -(8*n^2-42*n+40-(10*n-18)*x+2*x^2)*P(n-1,x) + (18*n-27+(20*n-39)*x+(2*n-13)*x^2-x^3)*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[(-(8*n^2 - 42*n + 40 - (10*n - 18)*x + 2*x^2)*P[n - 1, x] + (18*n - 27 + (20*n - 39)*x + (2*n - 13)*x^2 - x^3)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1 - (n - 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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