A222412 Denominators in Taylor series expansion of (x/(exp(x) - 1))^(3/2)*exp(x/2).

1, 4, 32, 384, 10240, 40960, 61931520, 49545216, 7927234560, 475634073600, 1993133260800, 177167400960, 48753634065776640, 195014536263106560, 39002907252621312000, 842462796656620339200, 2204424056667635712000, 79359266040034885632000

Offset: 0

Author

N. J. A. Sloane, Feb 28 2013

Keywords

• nonn
• frac

Examples

			The first few fractions are 1, -1/4, -1/32, 5/384, 7/10240, -19/40960, -869/61931520, 715/49545216, ... = A222411/A222412. - _Petros Hadjicostas_, May 14 2020
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.
• F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011.
• F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130.
• D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.

Crossrefs

Cf. A222411.

Cf. also A241885/A242225, A350194/A350154.

Programs

Maple

gf:= (x/(exp(x)-1))^(3/2)*exp(x/2):
a:= n-> denom(coeff(series(gf, x, n+3), x, n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 02 2013

Mathematica

Series[(x/(Exp[x]-1))^(3/2)*Exp[x/2], {x, 0, 25}] // CoefficientList[#, x]& // Denominator (* Jean-François Alcover, Mar 18 2014 *)

Formula

Theorem: A241885(n)/A242225(n) = n!*A222411(n)/(A222412(n)*(-1)^n/(1-2*n)) = n!*A350194(n)/(A350154(n)*(2*n+1)). - David Broadhurst, Apr 23 2022 (see Link).