A173175 - OEIS

%I #26 Jan 02 2019 10:48:04

%S 0,8,2400,1825200,2687489280,6503780163000,23436548406180000,

%T 117725514040791821024,786292024016459316676608,

%U 6739465778247681589030301160,72110357818535214970387726284000,942092946853627620313318842336862608,14758709413836719039368938494112056160000

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

%H Seiichi Manyama, <a href="/A173175/b173175.txt">Table of n, a(n) for n = 0..175</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F From _Seiichi Manyama_, Jan 02 2019: (Start)

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

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

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

%F a(n) ~ exp(1) * 2^(4*n - 2) * n^(2*n). - _Vaclav Kotesovec_, Jan 02 2019

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

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

%o (PARI) {a(n) = (polchebyshev(2*n, 1, 2*n+1)-1)/2} \\ _Seiichi Manyama_, Jan 02 2019

%o (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

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

%K nonn

%O 0,2

%A _Artur Jasinski_, Feb 11 2010

%E a(11)-a(12) from _Seiichi Manyama_, Jan 02 2019