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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":"85f23ca6f08d8b31b709b66bb8890919c46520534e55373379cbeb4c47a22112","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-96e4112733996073","role":"sequence","name":"veronese_section_2_1 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","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: 1;\n  2: 1;\n  3: 1, 1;\n  4: 2, 2;\n  5: 1, 3,  1;\n  6: 1, 7,  3;\n  7: 0, 6,  6,  1;\n  8: 0, 8, 16,  4;\n  9: 0, 2, 19, 10, 1;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = -(2-9*x+13*x^2)*P(n-6,x) - (x-5*x^2+4*x^3)*d/dx P(n-6,x) - (4-14*x+4*x^2)*P(n-4,x) - (2*x-8*x^2)*d/dx P(n-4,x) + (3+8*x)*P(n-2,x) + (1-4*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 1, P(3,x) = 1 + x, P(4,x) = 2 + 2*x, P(5,x) = 1 + 3*x + x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 1 + x;\nP[4, x_] := 2 + 2*x;\nP[5, x_] := 1 + 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2 - 9*x + 13*x^2)*P[n - 6, x] - (x - 5*x^2 + 4*x^3)*D[P[n - 6, x], x] - (4 - 14*x + 4*x^2)*P[n - 4, x] - (2*x - 8*x^2)*D[P[n - 4, x], x] + (3 + 8*x)*P[n - 2, x] + (1 - 4*x)*D[P[n - 2, x], x])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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