A336061 Numerators of coefficients associated with the second virial coefficient for rigid spheres with imbedded point dipoles.

1, 1, 29, 11, 13, 17, 523, 31, 66197, 83651, 21253, 3660541, 520783, 668861, 3322147, 30013913, 12938197, 4073039057, 310878307, 6867070733, 668207557, 104732138813, 56875471, 253267848881, 6285904022089, 913083596083, 2612577367192619, 3420422655984353

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.

Links

Table of n, a(n) for n=1..28.

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)).

A336061(n)/A336062(n) ~ exp(2*n) / (12*sqrt(Pi) * n^(2*n + 7/2)). - Vaclav Kotesovec, Jul 14 2020

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

Cf. A006134, A336062 (denominators).

Sequence in context: A309007 A070714 A040816 * A160494 A332940 A165769

Adjacent sequences: A336058 A336059 A336060 * A336062 A336063 A336064

Keyword

nonn,easy,frac

Author

Jan Mangaldan, Jul 07 2020

Status

approved