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A107270 Multiples of coefficients in asymptotic expansion of the rotational partition function for a heteronuclear diatomic molecule. 1

%I

%S 1,1,2,8,72,1440,55008,3507840,342679680,48401625600,9472057781760,

%T 2484361405532160,850218223244544000,371335242657899520000,

%U 203148791342840318976000,137006974339300359770112000

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

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

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

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

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

%F a(n) ~ 2^(n + 7/2) * n^(3*n + 3/2) / (exp(3*n) * Pi^(2*n - 1/2)). - _Vaclav Kotesovec_, Jun 08 2019

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

%t 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 *)

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

%Y Cf. A198954.

%K nonn

%O 0,3

%A _Michael Somos_, May 15 2005

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Last modified May 28 15:04 EDT 2022. Contains 354115 sequences. (Running on oeis4.)