|
%I #59 Feb 04 2022 13:31:53
%S 1,2,44,4332,1012664,432457640,293661065788,290711372717976,
%T 395320344293410544,707125993042984343136,1609908874238209683872480,
%U 4545914321591993313415189408,15591582457233317184439165505544,63847180690107503874880321918389332
%N a(n) is the constant term in the expansion of ( Sum_{k=1..n} x_k^(2*k-1) + x_k^(-(2*k-1)) )^(2*n).
%H Seiichi Manyama, <a href="/A329021/b329021.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = A077045(2*n).
%F a(n) = Sum_{k=0..floor((2*n-1)/2)} (-1)^k*binomial(2*n,k)*binomial((2*n+1)*n-2*n*k-1,(2*n-1)*n-2*n*k) for n > 0.
%F a(n) ~ sqrt(3) * 2^(2*n - 1) * n^(2*n - 3/2) / sqrt(Pi). - _Vaclav Kotesovec_, Feb 04 2022
%e (x + 1/x)^2 = x^2 + 2 + 1/x^2. So a(1) = 2.
%t a[0] = 1; a[n_] := Sum[(-1)^k * Binomial[2*n, k] * Binomial[(2*n + 1)*n - 2*n*k - 1, (2*n - 1)*n - 2*n*k], {k, 0, Floor[n - 1/2]}]; Array[a, 14, 0] (* _Amiram Eldar_, May 06 2021 *)
%o (PARI) {a(n) = polcoef((sum(k=1, n, x^(2*k-1)+x^(-(2*k-1))))^(2*n), 0)}
%o (PARI) {a(n) = if(n==0, 1, sum(k=0,(2*n-1)\2, (-1)^k*binomial(2*n, k)*binomial((2*n+1)*n-2*n*k-1, (2*n-1)*n-2*n*k)))}
%Y Main diagonal of A329020.
%Y Cf. A077045.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Nov 02 2019
|
