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A336061
Numerators of coefficients associated with the second virial coefficient for rigid spheres with imbedded point dipoles.
2
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.
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
KEYWORD
nonn,easy,frac
AUTHOR
Jan Mangaldan, Jul 07 2020
STATUS
approved