OFFSET
0,2
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..244
FORMULA
G.f.: hypergeom([1/7, 2/7, 3/7, 4/7, 5/7, 6/7], [1/2, 1/2, 1/2, 1, 1], (7^7)*x/2^6).
E.g.f.: hypergeom([1/7, 2/7, 3/7, 4/7, 5/7, 6/7], [1/2, 1/2, 1/2, 1, 1, 1], (7^7)*x/2^6).
a(n) = Integral_{x=0..7^7/2^6} x^n*W(x), where W(x) = (8*7^(1/2-7)/Pi^(3/2))*MeijerG([[],[0,0,0,-1/2,-1/2,-1/2]],[[-1/7, -2/7, -3/7, -4/7, -5/7, -6/7],[]],x/(7^7/2^6)), where MeijerG is the Meijer G-function (here in Maple notation).
W(x) is a positive function on x = (0, 7^7/2^6), it is singular at x = 0 and it decreases monotonically to zero at x = 7^7/2^6. W(x) can be represented in terms of six generalized hypergeometric functions of type 6F5. This integral representation is unique, as W(x) is the solution of the Hausdorff power moment problem.
a(n) ~ 7^(7*n+1/2)*8^(-2*n-1)/(n*Pi)^(3/2). - Stefano Spezia, Mar 27 2026
a(n) = binomial(7*n,2*n)*binomial(5*n,2*n)*binomial(3*n,n) = binomial(7*n,n)*binomial(6*n,2*n)*binomial(4*n,2*n). - Chai Wah Wu, Mar 28 2026
8*(2*n + 1)^3*(n + 1)^3*a(n + 1) = 7*(7*n + 6)*(7*n + 5)*(7*n + 4)*(7*n + 3)*(7*n + 2)*(7*n + 1)*a(n). - Robert Israel, Mar 30 2026
MATHEMATICA
A394642[n_] := Binomial[7*n, 2*n]*Binomial[5*n, 2*n]*Binomial[3*n, n];
Array[A394642, 15, 0] (* Paolo Xausa, Mar 29 2026 *)
PROG
(Python)
from math import comb
def A394642(n): return comb(7*n, 2*n)*comb(5*n, 2*n)*comb(3*n, n) # Chai Wah Wu, Mar 28 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Karol A. Penson, Mar 27 2026
STATUS
approved