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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":"239e4456b4aef96360e881f60ec690571e855a1e3e650e01d1a8adc023f5174d","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-c498b183c57d9f9c","role":"sequence","name":"veronese_section_2_0 of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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:   0;\n  2:   1;\n  3:   6;\n  4:  15,    2;\n  5:  28,   30;\n  6:  45,  140,    3;\n  7:  66,  420,   84;\n  8:  91,  990,  630,   4;\n  9: 120, 2002, 2772, 180;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12*x^2+x^3)*P(n-6,x) + 4*x^3*d/dx P(n-6,x) - (24*x+15*x^2)*P(n-5,x) - (12*x^2+4*x^3)*d/dx P(n-5,x) + (12+3*x-3*x^2)*P(n-4,x) + 12*x*d/dx P(n-4,x) + (27+10*x)*P(n-3,x) - (4+4*x-8*x^2)*d/dx P(n-3,x) - (11-3*x)*P(n-2,x) + (8+12*x)*d/dx P(n-2,x) + 5*P(n-1,x) - (4+4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 6, P(4,x) = 15 + 2*x, P(5,x) = 28 + 30*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 6;\nP[4, x_] := 15 + 2*x;\nP[5, x_] := 28 + 30*x;\nP[n_, x_] := P[n, x] = Expand[(12*x^2 + x^3)*P[n - 6, x] + 4*x^3*D[P[n - 6, x], x] - (24*x + 15*x^2)*P[n - 5, x] - (12*x^2 + 4*x^3)*D[P[n - 5, x], x] + (12 + 3*x - 3*x^2)*P[n - 4, x] + 12*x*D[P[n - 4, x], x] + (27 + 10*x)*P[n - 3, x] - (4 + 4*x - 8*x^2)*D[P[n - 3, x], x] - (11 - 3*x)*P[n - 2, x] + (8 + 12*x)*D[P[n - 2, x], x] + 5*P[n - 1, x] - (4 + 4*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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