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A361657 Constant term in the expansion of (1 + x^2 + y^2 + 1/(x*y))^n.

%I A361657 #21 Mar 20 2023 07:32:44
%S A361657 1,1,1,1,13,61,181,421,1261,5293,21421,73261,232321,789361,2954953,
%T A361657 11127481,39961741,139908301,499315501,1835933293,6792310153,
%U A361657 24827506873,90058277233,328509505633,1210097040769,4473191880961,16495696956961,60721903812961
%N A361657 Constant term in the expansion of (1 + x^2 + y^2 + 1/(x*y))^n.
%F A361657 a(n) = n! * Sum_{k=0..floor(n/4)} 1/(k!^2 * (2*k)! * (n-4*k)!) = Sum_{k=0..floor(n/4)} binomial(n,4*k) * A000897(k).
%F A361657 From _Vaclav Kotesovec_, Mar 20 2023: (Start)
%F A361657 Recurrence: (n-2)*n^2*a(n) = (4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - (n-1)*(6*n^2 - 18*n + 13)*a(n-2) + 4*(n-2)^2*(n-1)*a(n-3) + 63*(n-3)*(n-2)*(n-1)*a(n-4).
%F A361657 a(n) ~ (1 + 2*sqrt(2))^(n+1) / (4*Pi*n). (End)
%t A361657 Table[n!*Sum[1/(k!^2*(2*k)!*(n - 4*k)!), {k, 0, n/4}], {n, 0, 30}] (* _Vaclav Kotesovec_, Mar 20 2023 *)
%o A361657 (PARI) a(n) = n!*sum(k=0, n\4, 1/(k!^2*(2*k)!*(n-4*k)!));
%Y A361657 Cf. A002426, A361658.
%Y A361657 Cf. A000897, A344560, A361637.
%K A361657 nonn
%O A361657 0,5
%A A361657 _Seiichi Manyama_, Mar 19 2023