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 A173194 a(n) = -sin^2 (2*n*arccos n) = - sin^2 (2*n*arcsin n). 2
 0, 0, 9408, 384199200, 54471499791360, 20405558846592060000, 16793517249722147195701440, 26730228454204365035835498694848, 75019085697452515216001640927169855488, 346154755746154620929434271983392498083891520 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..107 Wikipedia, Chebyshev polynomials. FORMULA 4*a(n) = ( (n+sqrt(n^2-1))^(2*n) - (n-sqrt(n^2-1))^(2*n) )^2. - Artur Jasinski, Feb 17 2010 From Seiichi Manyama, Jan 05 2019: (Start) a(n) = (n^2-1) * n^2 * (Sum_{k=0..n-1} binomial(2*n,2*k+1)*(n^2-1)^(n-1-k)*n^(2*k))^2. For n > 0, a(n) = (n^2-1) * U_{2*n-1}(n)^2 where U_{n}(x) is a Chebyshev polynomial of the second kind. (End) a(n) ~ 2^(4*n - 2) * n^(4*n). - Vaclav Kotesovec, Jan 05 2019 MAPLE A173194 := proc(n) ((n+sqrt(n^2-1))^(2*n)-(n-sqrt(n^2-1))^(2*n))^2 ; expand(%/4) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011 MATHEMATICA Round[Table[ -N[Sin[2 n ArcSin[n]], 100]^2, {n, 0, 15}]] (* Artur Jasinski *) Table[FullSimplify[(-1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x))^2], {x, 0, 7}] (* Artur Jasinski, Feb 17 2010 *) Table[(n^2-1)*ChebyshevU[2*n-1, n]^2, {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *) PROG (PARI) {a(n) = (n^2-1)*n^2*(sum(k=0, n-1, binomial(2*n, 2*k+1)*(n^2-1)^(n-1-k)*n^(2*k)))^2} \\ Seiichi Manyama, Jan 05 2019 (PARI) {a(n) = (n^2-1)*polchebyshev(2*n-1, 2, n)^2} \\ Seiichi Manyama, Jan 05 2019 CROSSREFS Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A173174, A173175, A173176. Sequence in context: A330838 A064587 A275922 * A239609 A231763 A206251 Adjacent sequences:  A173191 A173192 A173193 * A173195 A173196 A173197 KEYWORD nonn AUTHOR Artur Jasinski, Feb 12 2010 EXTENSIONS a(9) from Seiichi Manyama, Jan 05 2019 STATUS approved

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Last modified October 30 12:22 EDT 2020. Contains 338079 sequences. (Running on oeis4.)