A361704 Constant term in the expansion of (1 + w^2 + x^2 + y^2 + z^2 + 1/(w*x*y*z))^n.

1, 1, 1, 1, 1, 1, 361, 2521, 10081, 30241, 75601, 166321, 1580041, 16833961, 114594481, 569368801, 2273150881, 7723366561, 30024671041, 193227592321, 1460787267601, 9492136169041, 50996729017081, 232560967743721, 973251617544361, 4464217099881001

Offset

0,7

Links

Table of n, a(n) for n=0..25.

Formula

a(n) = Sum_{k=0..floor(n/6)} (4*k)!/k!^4 * binomial(6*k,4*k) * binomial(n,6*k).

From Vaclav Kotesovec, Mar 25 2023: (Start)

Recurrence: (n-3)*n^4*a(n) = (6*n^5 - 30*n^4 + 50*n^3 - 45*n^2 + 21*n - 4)*a(n-1) - (n-1)*(15*n^4 - 90*n^3 + 195*n^2 - 195*n + 76)*a(n-2) + 5*(n-2)*(n-1)*(4*n^3 - 24*n^2 + 48*n - 33)*a(n-3) - 5*(n-3)*(n-2)*(n-1)*(3*n^2 - 15*n + 19)*a(n-4) + 6*(n-4)*(n-3)^2*(n-2)*(n-1)*a(n-5) + 11663*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).

a(n) ~ sqrt(7 + 9/(4*2^(1/3)) + 433/(48*2^(2/3))) * (1 + 3*2^(2/3))^n / (Pi^2 * n^2). (End)

Mathematica

Table[Sum[(4*k)!/k!^4 * Binomial[6*k, 4*k] * Binomial[n, 6*k], {k, 0, n/6}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 25 2023 *)

Prog

(PARI) a(n) = sum(k=0, n\6, (4*k)!/k!^4*binomial(6*k, 4*k)*binomial(n, 6*k));

Crossrefs

Cf. A361675, A361703, A361705.

Sequence in context: A211822 A233552 A176899 * A302229 A302672 A302476

Adjacent sequences: A361701 A361702 A361703 * A361705 A361706 A361707

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 21 2023

Status

approved