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

1, 1, 1, 1, 13, 61, 181, 421, 1261, 5293, 21421, 73261, 232321, 789361, 2954953, 11127481, 39961741, 139908301, 499315501, 1835933293, 6792310153, 24827506873, 90058277233, 328509505633, 1210097040769, 4473191880961, 16495696956961, 60721903812961

Offset

0,5

Links

Table of n, a(n) for n=0..27.

Formula

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).

From Vaclav Kotesovec, Mar 20 2023: (Start)

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).

a(n) ~ (1 + 2*sqrt(2))^(n+1) / (4*Pi*n). (End)

Mathematica

Table[n!*Sum[1/(k!^2*(2*k)!*(n - 4*k)!), {k, 0, n/4}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 20 2023 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\4, 1/(k!^2*(2*k)!*(n-4*k)!));

Crossrefs

Cf. A002426, A361658.

Cf. A000897, A344560, A361637.

Sequence in context: A270449 A139880 A127876 * A308461 A357964 A353223

Adjacent sequences: A361654 A361655 A361656 * A361658 A361659 A361660

Keyword

nonn

Author

Seiichi Manyama, Mar 19 2023

Status

approved