OFFSET
0,2
FORMULA
G.f.: hypergeom([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8], [1/3, 2/3, 1, 1, 1, 1], (8^8*x)/3^3).
E.g.f.: hypergeom([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8], [1/3, 2/3, 1, 1, 1, 1, 1], (8^8*x)/3^3).
a(n) = Integral_{x=0..8^8/3^3} x^n*W(x), where W(x) = ((9*sqrt(3))/(2*(8^8)*Pi^(5/2)))*MeijerG([[],[0,0,0,0,0,-1/3,-2/3]],[[-1/8, -1/4, -3/8, -1/2, -5/8, -3/4, -7/8],[]],x/(8^8/3^3)), where MeijerG is the Meijer G-function (here in Maple notation). W(x) is a positive function on x=(0, 8^8/3^3), it is singular at x = 0 and it decreases monotonically to zero at x = 8^8/3^3. W(x) can be represented in terms of seven generalized hypergeometric functions of type 7F6. This integral representation is unique, as W(x) is the solution of the Hausdorff power moment problem.
a(n) ~ 2^(24*n-1)*3^(-3*n-1/2)/(n*Pi)^(5/2). - Stefano Spezia, Feb 13 2026
a(n) = binomial(8*n,3*n)*binomial(5*n,n)*binomial(4*n,n)*binomial(3*n,n)*binomial(2*n,n). - Chai Wah Wu, Feb 14 2026
D-finite with recurrence 3*n^5*(3*n-1)*(3*n-2)*a(n) -128*(8*n-5)*(4*n-1)*(8*n-7)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n-1)=0. - R. J. Mathar, Feb 24 2026
a(n) = binomial(8*n,n)*binomial(7*n,n)*binomial(6*n,n)*binomial(5*n,n)*binomial(4*n,n) = binomial(8*n,n)*binomial(7*n,3*n)*binomial(4*n,n)*binomial(3*n,n)*binomial(2*n,n) = binomial(8*n,n)*binomial(7*n,n)*binomial(6*n,3*n)*binomial(3*n,n)*binomial(2*n,n) = binomial(8*n,n)*binomial(7*n,n)*binomial(6*n,n)*binomial(5*n,2*n)*binomial(2*n,n). - Chai Wah Wu, Mar 31 2026
MATHEMATICA
a[n_]:=(8*n)!/((3*n)!*(n!)^5); Array[a, 11, 0] (* Stefano Spezia, Feb 13 2026 *)
PROG
(Python)
from math import comb, prod
def A393378(n): return comb(8*n, 3*n)*prod(comb(k*n, n) for k in range(2, 6)) # Chai Wah Wu, Feb 14 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Feb 13 2026
STATUS
approved