A361677 Constant term in the expansion of (1 + x + y + z + 1/(x*y) + 1/(y*z) + 1/(z*x))^n.

1, 1, 1, 19, 73, 181, 1711, 10081, 38809, 256033, 1696861, 8388271, 49449511, 326195299, 1847392093, 10789655059, 69202030969, 418647580489, 2498113460881, 15735859252147, 97919649290053, 598317173139313, 3748943081117323

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = Sum_{k=0..floor(n/3)} (3*k)!/k!^3 * binomial(3*k,k) * binomial(n,3*k).

From Vaclav Kotesovec, Mar 22 2023: (Start)

Recurrence: 2*n^3*(2*n - 3)*a(n) = 2*(10*n^4 - 32*n^3 + 38*n^2 - 22*n + 5)*a(n-1) - 2*(n-1)*(2*n - 3)*(10*n^2 - 24*n + 17)*a(n-2) + (n-2)*(n-1)*(769*n^2 - 2331*n + 1594)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*(739*n - 1481)*a(n-4) + 733*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).

a(n) ~ sqrt(733/108 + 1/2^(2/3) + 9/2^(4/3)) * (1 + 9/2^(2/3))^n / (2 * Pi^(3/2) * n^(3/2)). (End)

Mathematica

Table[Sum[(3*k)!/k!^3 * Binomial[3*k, k] * Binomial[n, 3*k], {k, 0, n/3}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 22 2023 *)

Prog

(PARI) a(n) = sum(k=0, n\3, (3*k)!/k!^3*binomial(3*k, k)*binomial(n, 3*k));

Crossrefs

Cf. A201805, A361678.

Cf. A275047, A344560.

Sequence in context: A141960 A178541 A157889 * A255897 A220447 A294460

Adjacent sequences: A361674 A361675 A361676 * A361678 A361679 A361680

Keyword

nonn

Author

Seiichi Manyama, Mar 20 2023

Status

approved