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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":"5495be9120bd32f8bdc872be06ac893f75cc152e12f5c7099015d52cbd1129cb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3a7609bd7546c802","role":"sequence","name":"Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:    1;\n   2:    2,      4;\n   3:  -13,     16,      16;\n   4:   36,   -152,      96,       64;\n   5:  -75,    592,   -1200,      512,      256;\n   6:  134,  -1652,    6016,    -7936,     2560,      1024;\n   7: -217,   3808,  -21024,    48640,   -47360,     12288,     4096;\n   8:  328,  -7728,   59328,  -204160,   343040,   -264192,    57344,    16384;\n   9: -471,  14304, -144672,   681472, -1680640,   2207744, -1404928,   262144,   65536;\n  10:  650, -24684,  316800, -1940224,  6476288, -12364800, 13303808, -7208960, 1179648, 262144;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-6*n^2+n+3+(8*n-8)*x-(2*n-2)*x^2)*P(n,x) = -(2*n^3-5*n-(4*n^3-14*n)*x-14*n*x^2+4*n*x^3)*P(n-1,x) - ((4*n^2-4*n-2)*x-(4*n^2-4*n-10)*x^2-12*x^3+4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 5*n - (4*n^3 - 14*n)*x - 14*n*x^2 + 4*n*x^3)*P[n - 1, x] - ((4*n^2 - 4*n - 2)*x - (4*n^2 - 4*n - 10)*x^2 - 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + n + 3 + (8*n - 8)*x - (2*n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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