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a(n) = T_{2*n}(n+1) where T_{n}(x) is a Chebyshev polynomial of the first kind.
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%I #19 Apr 15 2020 05:27:07

%S 1,7,577,119071,46099201,28860511751,26650854921601,34100354867927167,

%T 57780789062419261441,125283240358674708816199,

%U 338393251269110482793304001,1114259437504123772777608493087,4394174409561746573589926449440001

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

%H Seiichi Manyama, <a href="/A322899/b322899.txt">Table of n, a(n) for n = 0..193</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 a(n) = T_{n}(2*n^2+4*n+1).

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

%F a(n) ~ exp(2) * 2^(2*n-1) * n^(2*n). - _Vaclav Kotesovec_, Apr 15 2020

%t a[n_] := ChebyshevT[2n, n+1];

%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Jan 02 2019 *)

%o (PARI) {a(n) = polchebyshev(2*n, 1, n+1)}

%o (PARI) {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n^2+2*n)^(n-k)*(n+1)^(2*k))}

%Y Diagonal of A188644.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 30 2018