A107270 Multiples of coefficients in asymptotic expansion of the rotational partition function for a heteronuclear diatomic molecule.

1, 1, 2, 8, 72, 1440, 55008, 3507840, 342679680, 48401625600, 9472057781760, 2484361405532160, 850218223244544000, 371335242657899520000, 203148791342840318976000, 137006974339300359770112000

Offset

0,3

References

G. Herzberg, Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules, D. Van Nostrand, 1945. see page 505

D. A. McQuarrie, Statistical Mechanics, University Science Books, 2000, see page 100 equ. (6-35)

G. H. Wannier, Statistical Physics, Dover Publications, 1987. see page 216 equ. (11.21)

Links

Table of n, a(n) for n=0..15.

Formula

Sum_{k>=0} (2*k + 1) * exp(-x*(k^2 + k)) ~ (1/x) * Sum_{k>=0} a(k) * (2*x)^k / (2*k + 1)!.

a(n) ~ 2^(n + 7/2) * n^(3*n + 3/2) / (exp(3*n) * Pi^(2*n - 1/2)). - Vaclav Kotesovec, Jun 08 2019

Example

1 + 3*exp(-2*x) + 5*exp(-6*x) + 7*exp(-10*x) + ... ~ 1/x + 1/3 + (1/15)*x + (4/315)*x^2 + ...

Mathematica

a[ n_] := If[ n < 0, 0, Sum[ BernoulliB[n + j] / (j! (n - j)!), {j, 0, n }] (2 n + 1)! / (-2)^n]; (* Michael Somos, Dec 04 2013 *)

Prog

(PARI) {a(n) = if( n<0, 0, sum( j=0, n, bernfrac(n+j) / ((n-j)! * j!)) * (2*n + 1)! / (-2)^n)};

Crossrefs

Cf. A198954.

Sequence in context: A194499 A009478 A038057 * A294351 A342177 A125814

Adjacent sequences: A107267 A107268 A107269 * A107271 A107272 A107273

Keyword

nonn

Author

Michael Somos, May 15 2005

Status

approved