A173175 a(n) = sinh^2( 2n*arcsinh(sqrt n)).

0, 8, 2400, 1825200, 2687489280, 6503780163000, 23436548406180000, 117725514040791821024, 786292024016459316676608, 6739465778247681589030301160, 72110357818535214970387726284000, 942092946853627620313318842336862608, 14758709413836719039368938494112056160000

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..175

Wikipedia, Chebyshev polynomials.

Index entries for sequences related to Chebyshev polynomials.

Formula

From Seiichi Manyama, Jan 02 2019: (Start)

a(n) = A322699(n,2*n).

a(n) = (T_{2*n}(2*n+1) - 1)/2 where T_{n}(x) is a Chebyshev polynomial of the first kind.

a(n) = 1/2 * (-1 + Sum_{k=0..2*n} binomial(4*n,2*k)*(n+1)^(2*n-k)*n^k). (End)

a(n) ~ exp(1) * 2^(4*n - 2) * n^(2*n). - Vaclav Kotesovec, Jan 02 2019

Maple

A173175 := proc(n) sinh(2*n*arcsinh(sqrt(n))) ; %^2 ; expand(%); simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011

Mathematica

Table[Round[N[Sinh[(2 n) ArcSinh[Sqrt[n]]]^2, 100]], {n, 0, 20}]

Prog

(PARI) {a(n) = (polchebyshev(2*n, 1, 2*n+1)-1)/2} \\ Seiichi Manyama, Jan 02 2019
(PARI) {a(n) = 1/2*(-1+sum(k=0, 2*n, binomial(4*n, 2*k)*(n+1)^(2*n-k)*n^k))} \\ Seiichi Manyama, Jan 02 2019

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A322699.

Sequence in context: A303101 A302952 A151580 * A268150 A325062 A247733

Adjacent sequences: A173172 A173173 A173174 * A173176 A173177 A173178

Keyword

nonn

Author

Artur Jasinski, Feb 11 2010

Extensions

a(11)-a(12) from Seiichi Manyama, Jan 02 2019

Status

approved