OFFSET
1,3
REFERENCES
J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., 1964, pages 210-211.
FORMULA
a(n) = numerator(1/(8 * Pi * (2*n)! * (2*n - 1)) * Integral_{w=0..2*Pi} Integral_{v=0..Pi} Integral_{u=0..Pi} (2 * cos(u) * cos(v) - sin(u) * sin(v) * cos(w))^(2 * n) * sin(u) * sin(v)).
a(n) = numerator(4^n * hypergeom([1, -n], [1/2 - n], 1/4)/((2 * n)! (2 * n - 1) (2 * n + 1)^2)).
a(n) = numerator(4^n*(Sum_{j=0..n} binomial(2*j,j))/(binomial(2*n,n)*(2*n)!*(2*n-1)*(2*n+1)^2)).
EXAMPLE
1/3, 1/75, 29/55125, 11/694575, 13/36018675, 17/2678348673, 523/5934977173125, ...
MATHEMATICA
Table[Numerator[4^k Sum[Binomial[2 j, j]/Binomial[2 k, k], {j, 0, k}]/((2 k)! (2 k - 1) (2 k + 1)^2)], {k, 20}]
Table[Numerator[4^k Hypergeometric2F1[1, -k, 1/2 - k, 1/4]/((2 k)! (2 k - 1) (2 k + 1)^2)], {k, 20}]
PROG
(PARI) a(n)={numerator(4^n*sum(j=0, n, binomial(2*j, j))/(binomial(2*n, n)*(2*n)!*(2*n-1)*(2*n+1)^2))} \\ Andrew Howroyd, Jul 07 2020
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Jan Mangaldan, Jul 07 2020
STATUS
approved