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Denominators of coefficients associated with the second virial coefficient for rigid spheres with imbedded point dipoles.
2

%I #16 Jul 14 2020 06:53:51

%S 3,75,55125,694575,36018675,2678348673,5934977173125,31414073315625,

%T 7287392748056045625,1197275761489443260625,46668548892583246253625,

%U 1437557979280466067633984375,42189201565839765028388671875,12773202666073647259994954296875,16951256433371736928038065776171875

%N Denominators 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) = 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)).

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

%F 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)).

%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[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}]

%t Table[Denominator[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)={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

%Y Cf. A006134, A336061 (numerators).

%K nonn,easy,frac

%O 1,1

%A _Jan Mangaldan_, Jul 07 2020