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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":"10bfa1380ffe9d23d481f2df77471a5d2c4c35b85f97a13d125ec9638d42c1f3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-3fea9ad1525b48eb","role":"sequence","name":"Triangle of Ward numbers T(n,k) read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    0;\n   2:    3;\n   3:   10,     30;\n   4:   25,    210,      315;\n   5:   56,    980,     3780,      3780;\n   6:  119,   3836,    28350,     69300,      51975;\n   7:  246,  13650,   170940,    762300,    1351350,     810810;\n   8:  501,  45870,   908985,   6546540,   20495475,   28378350,    14189175;\n   9: 1012, 148632,  4461600,  48488440,  237537300,  567567000,   643242600,   275675400;\n  10: 2035, 470184, 20744724, 325725400, 2334382050, 8533685160, 16402686300, 15713497800, 5892561675;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2+4*n+2)*x+(n^3+2*n^2+n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2+2*n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+(2*n^2+4*n)*x)*P(n-1,x) + (n*x+n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 + 4*n + 2)*x + (n^3 + 2*n^2 + n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 + 2*n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + (2*n^2 + 4*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, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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