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User: Jan Mangaldan

Jan Mangaldan has authored 2 sequences.

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

Jan Mangaldan, Jul 07 2020

			1/3, 1/75, 29/55125, 11/694575, 13/36018675, 17/2678348673, 523/5934977173125, ...

J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., 1964, pages 210-211.

Cf. A006134, A336062 (denominators).

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}]

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

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

A336062 Denominators of coefficients associated with the second virial coefficient for rigid spheres with imbedded point dipoles.

Original entry on oeis.org

3, 75, 55125, 694575, 36018675, 2678348673, 5934977173125, 31414073315625, 7287392748056045625, 1197275761489443260625, 46668548892583246253625, 1437557979280466067633984375, 42189201565839765028388671875, 12773202666073647259994954296875, 16951256433371736928038065776171875

Offset: 1

Jan Mangaldan, Jul 07 2020

			1/3, 1/75, 29/55125, 11/694575, 13/36018675, 17/2678348673, 523/5934977173125, ...

J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., 1964, pages 210-211.

Cf. A006134, A336061 (numerators).

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}]

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

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

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User: Jan Mangaldan

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

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