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 A107270 Multiples of coefficients in asymptotic expansion of the rotational partition function for a heteronuclear diatomic molecule. 1
 1, 1, 2, 8, 72, 1440, 55008, 3507840, 342679680, 48401625600, 9472057781760, 2484361405532160, 850218223244544000, 371335242657899520000, 203148791342840318976000, 137006974339300359770112000 (list; graph; refs; listen; history; text; internal format)
 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 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 A125814 A295044 Adjacent sequences:  A107267 A107268 A107269 * A107271 A107272 A107273 KEYWORD nonn AUTHOR Michael Somos, May 15 2005 STATUS approved

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Last modified September 29 17:04 EDT 2020. Contains 337432 sequences. (Running on oeis4.)