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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":"8e6dbda494fa4157b31425694048ad6604c781293804af13c21edd76cde367c5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2872b2f2cd7c5725","role":"sequence","name":"Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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:    5,      2;\n  3:   14,     28,       3;\n  4:   33,    178,      99,       4;\n  5:   72,    826,    1239,     288,       5;\n  6:  151,   3264,   10179,    6528,     755,       6;\n  7:  310,  11748,   66435,   88580,   29370,    1860,      7;\n  8:  629,  39886,  376368,  899372,  627280,  119658,   4403,     8;\n  9: 1268, 130342, 1941897, 7596384, 9495480, 3883794, 456197, 10144, 9;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n+1)*x^2*P(n-4,x) - ((2*n+2)*x+(2*n^2-2*n-3)*x^2)*P(n-3,x) - ((n+1)*x^2-(n+1)*x^3)*d/dx P(n-3,x) - (2*n+2+(2*n^2-4*n-4)*x+(n^3-2*n^2)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-n-1)*x^2-n^2*x^3)*d/dx P(n-2,x) + (3*n+1+(2*n^2-2*n)*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x, P(3,x) = 14 + 28*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 5 + 2*x;\nP[3, x_] := 14 + 28*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1)*x^2*P[n - 4, x] - ((2*n + 2)*x + (2*n^2 - 2*n - 3)*x^2)*P[n - 3, x] - ((n + 1)*x^2 - (n + 1)*x^3)*D[P[n - 3, x], x] - (2*n + 2 + (2*n^2 - 4*n - 4)*x + (n^3 - 2*n^2)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - n - 1)*x^2 - n^2*x^3)*D[P[n - 2, x], x] + (3*n + 1 + (2*n^2 - 2*n)*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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