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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":"0d843a820f38b56a6deb707be50e129d05cbdd5de261fc68a84ee8d6b55095be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7466465df6cec0d5","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","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:       1;\n   2:       2,        1;\n   3:       8,        9,         1;\n   4:      44,       79,        21,         1;\n   5:     260,      741,       343,        38,        1;\n   6:    1556,     7375,      5265,      1005,       60,       1;\n   7:    9332,    76461,     79975,     23230,     2345,      87,      1;\n   8:   55988,   813079,   1221801,    505925,    76440,    4718,    119,     1;\n   9:  335924,  8775909,  18847711,  10721886,  2268525,  206514,   8554,   156,   1;\n  10: 2015540, 95527231, 293459313, 224503693, 63857556, 8024835, 484386, 14358, 198, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(6-3*x-4*x^2)*P(n-2,x) - (5*x-20*x^2)*d/dx P(n-2,x) + (7-3*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(6 - 3*x - 4*x^2)*P[n - 2, x] - (5*x - 20*x^2)*D[P[n - 2, x], x] + (7 - 3*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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