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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":"5bf9ff286b7a48c698f3342545f49ff304bc1d0717f3b5ba0af59dd5971d7885","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-cf38558d99f4fe37","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","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;\n  2:   7,    4;\n  3:  20,   16;\n  4:  45,   72,    16;\n  5:  86,  272,    96;\n  6: 147,  812,   528,    64;\n  7: 232, 2016,  2432,   512;\n  8: 345, 4368,  8928,  3328,  256;\n  9: 490, 8544, 27072, 17920, 2560;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-6*n^2+7*n+(8*n^2-22*n)*x+8*n*x^2)*P(n,x) = (2*n^3+n+3+(8*n^3-24*n^2+14*n-26)*x+(32*n^2-80*n+64)*x^2+(32*n-32)*x^3)*P(n-1,x) + ((4*n^2-4*n+2)*x-(16*n^2-32*n+20)*x^2-(64*n-64)*x^3-64*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + n + 3 + (8*n^3 - 24*n^2 + 14*n - 26)*x + (32*n^2 - 80*n + 64)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] + ((4*n^2 - 4*n + 2)*x - (16*n^2 - 32*n + 20)*x^2 - (64*n - 64)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 7*n + (8*n^2 - 22*n)*x + 8*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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