A336061 - OEIS

%I #18 Jul 14 2020 06:54:00

%S 1,1,29,11,13,17,523,31,66197,83651,21253,3660541,520783,668861,

%T 3322147,30013913,12938197,4073039057,310878307,6867070733,668207557,

%U 104732138813,56875471,253267848881,6285904022089,913083596083,2612577367192619,3420422655984353

%N Numerators of coefficients associated with the second virial coefficient for rigid spheres with imbedded point dipoles.

%D J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., 1964, pages 210-211.

%F 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)).

%F a(n) = numerator(4^n * hypergeom([1, -n], [1/2 - n], 1/4)/((2 * n)! (2 * n - 1) (2 * n + 1)^2)).

%F 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)).

%F A336061(n)/A336062(n) ~ exp(2*n) / (12*sqrt(Pi) * n^(2*n + 7/2)). - _Vaclav Kotesovec_, Jul 14 2020

%e 1/3, 1/75, 29/55125, 11/694575, 13/36018675, 17/2678348673, 523/5934977173125, ...

%t 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}]

%t Table[Numerator[4^k Hypergeometric2F1[1, -k, 1/2 - k, 1/4]/((2 k)! (2 k - 1) (2 k + 1)^2)], {k, 20}]

%o (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

%Y Cf. A006134, A336062 (denominators).

%K nonn,easy,frac

%O 1,3

%A _Jan Mangaldan_, Jul 07 2020