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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":"1b0739e74c8f7b0371db05f9f5478ea45af54ad9bb56edead2e86a436e3fb6fa","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-e655e253465c7aff","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.","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,  2;\n  1:  2,  2;\n  2:  1,  6,   9,   4;\n  3:  4, 12,  12,   4;\n  4:  1, 10,  30,  40,   25,    6;\n  5:  6, 30,  60,  60,   30,    6;\n  6:  1, 14,  63, 140,  175,  126,  49,   8;\n  7:  8, 56, 168, 280,  280,  168,  56,   8;\n  8:  1, 18, 108, 336,  630,  756, 588, 288, 81, 10;\n  9: 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n)*P(n,x) = -(n^3-3*n-(3*n^2+5*n)*x-(n^3+3*n^2+2*n)*x^2)*P(n-1,x) + (n^2+n-1+(n^2-n-4)*x-(n^2+5*n+5)*x^2-(n^2+3*n+2)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n - (3*n^2 + 5*n)*x - (n^3 + 3*n^2 + 2*n)*x^2)*P[n - 1, x] + (n^2 + n - 1 + (n^2 - n - 4)*x - (n^2 + 5*n + 5)*x^2 - (n^2 + 3*n + 2)*x^3)*D[P[n - 1, x], x])/(n^2 + n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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