cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

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%I A336061 #18 Jul 14 2020 06:54:00
%S A336061 1,1,29,11,13,17,523,31,66197,83651,21253,3660541,520783,668861,
%T A336061 3322147,30013913,12938197,4073039057,310878307,6867070733,668207557,
%U A336061 104732138813,56875471,253267848881,6285904022089,913083596083,2612577367192619,3420422655984353
%N A336061 Numerators of coefficients associated with the second virial coefficient for rigid spheres with imbedded point dipoles.
%D A336061 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 A336061 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 A336061 a(n) = numerator(4^n * hypergeom([1, -n], [1/2 - n], 1/4)/((2 * n)! (2 * n - 1) (2 * n + 1)^2)).
%F A336061 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 A336061(n)/A336062(n) ~ exp(2*n) / (12*sqrt(Pi) * n^(2*n + 7/2)). - _Vaclav Kotesovec_, Jul 14 2020
%e A336061 1/3, 1/75, 29/55125, 11/694575, 13/36018675, 17/2678348673, 523/5934977173125, ...
%t A336061 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 A336061 Table[Numerator[4^k Hypergeometric2F1[1, -k, 1/2 - k, 1/4]/((2 k)! (2 k - 1) (2 k + 1)^2)], {k, 20}]
%o A336061 (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 A336061 Cf. A006134, A336062 (denominators).
%K A336061 nonn,easy,frac
%O A336061 1,3
%A A336061 _Jan Mangaldan_, Jul 07 2020