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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":"f74ca2dae7889b6ef34c61b85bcb3d13b3f6fb3a3f5518e700a7e18546346a8a","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-62c0157c67ca5559","role":"sequence","name":"theta_plus_one of Triangle read by rows: 2-Stirling numbers of the second kind.","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:     2;\n   3:    10,      6;\n   4:    38,     54,      12;\n   5:   130,    330,     168,      20;\n   6:   422,   1710,    1500,     400,      30;\n   7:  1330,   8106,   10920,    4900,     810,     42;\n   8:  4118,  36414,   70812,   47600,   13020,   1470,    56;\n   9: 12610, 158010,  426888,  403620,  162540,  29988,  2464,   72;\n  10: 38342, 669870, 2450460, 3133200, 1743210, 467460, 62160, 3888, 90;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+3*x+x^3)*P(n-3,x) - (2*x+3*x^2)*d/dx P(n-3,x) - (x^2+x^3)*d^2/dx^2 P(n-3,x) - (11+6*x+3*x^2)*P(n-2,x) + 2*x*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 2, P(3,x) = 10 + 6*x.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[3, x_] := 10 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(6 + 3*x + x^3)*P[n - 3, x] - (2*x + 3*x^2)*D[P[n - 3, x], x] - (x^2 + x^3)*D[P[n - 3, x], {x, 2}] - (11 + 6*x + 3*x^2)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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