|
|
A336062
|
|
Denominators of coefficients associated with the second virial coefficient for rigid spheres with imbedded point dipoles.
|
|
2
|
|
|
3, 75, 55125, 694575, 36018675, 2678348673, 5934977173125, 31414073315625, 7287392748056045625, 1197275761489443260625, 46668548892583246253625, 1437557979280466067633984375, 42189201565839765028388671875, 12773202666073647259994954296875, 16951256433371736928038065776171875
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
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
|
|
|
FORMULA
|
a(n) = denominator(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) = denominator(4^n * hypergeom([1, -n], [1/2 - n], 1/4)/((2 * n)! (2 * n - 1) (2 * n + 1)^2)).
a(n) = denominator(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[Denominator[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[Denominator[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)={denominator(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
|
|
|
STATUS
|
approved
|
|
|
|