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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":"8057b06f6dc4169bc1e049533545606463bbd6394377250d76d65a64854e9c25","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-6d092678b88e72ae","role":"sequence","name":"theta_plus_one of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","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,    2;\n  2: 2,    8,     3;\n  3: 2,   20,    21,      4;\n  4: 2,   44,    93,     44,      5;\n  5: 2,   92,   345,    300,     80,      6;\n  6: 2,  188,  1173,   1660,    775,    132,     7;\n  7: 2,  380,  3801,   8204,   5950,   1722,   203,    8;\n  8: 2,  764, 11973,  37884,  40005,  17472,  3430,  296,   9;\n  9: 2, 1532, 37065, 167500, 247380, 152838, 44394, 6288, 414, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(x^2-x^3)*P(n-3,x) - (x^2-x^3)*d/dx P(n-3,x) + (1+x-3*x^2)*P(n-2,x) + (x-2*x^2)*d/dx P(n-2,x) + 3*x*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 2 + 8*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 2 + 8*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(x^2 - x^3)*P[n - 3, x] - (x^2 - x^3)*D[P[n - 3, x], x] + (1 + x - 3*x^2)*P[n - 2, x] + (x - 2*x^2)*D[P[n - 2, x], x] + 3*x*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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