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%I #10 Aug 30 2019 03:34:09
%S 1,1,1,1,1,2,1,1,2,1,1,2,4,2,8,1,5,4,1,8,4,1,1,2,2,8,8,16,1,7,14,1,8,
%T 4,16,2,1,4,56,4,16,32,64,16,128,1,3,8,6,16,8,64,8,128,64,1,5,2,12,16,
%U 16,160,16,128,128,256,1,11,22,33,176,8,32,4,128,64,256,64,1,2,44,22,88,32
%N Table of numerators of coefficients of certain rational polynomials.
%C These rational polynomials R(n;x) appear in the evaluation of an integral in thermal field theories in the Bose case. See the Haber and Weldon reference eq. (D1), l. 2, p. 1857 and the W. Lang link.
%D H. E. Haber and H. A. Weldon, On the relativistic Bose-Einstein integrals, J. Math. Phys. 23(10) (1982) 1852-1858.
%H W. Lang: <a href="/A101022/a101022.txt">Rational polynomials R(n,x)</a>
%F a(n, m)= numerator(R(n, x)[x^m]), m=0, ..., n-1, n>=1, with the rational polynomials R(n, x) of degree n-1 defined by R(n, x):=hypergeom([1, 1, 1-n], [3/2, 2], -x/2)) = sum(R(n, m)*x^m, m=0..n-1), n>=1.
%F The rational polynomials are R(n, x) = 1 + sum(binomial(n-1, m)/((m+1)*(2*m+1)*binomial(2*m, m))*(2*x)^m, m=1..n-1), n>=1.
%F a(n, m)=numerator(R(n, m)) with R(n, m) = binomial(n-1, m)/((m+1)*(2*m+1)*binomial(2*m, m))*2^m, m=1..n-1, n=1, 2, ... and R(n, 0)=1, n>=1, else 0.
%e The rows of the rational table are: [1/1]; [1/1, 1/6]; [1/1, 1/3, 2/45]; [1/1, 1/2, 2/15, 1/70]; ...
%Y The table of denominators is given in A101023.
%K nonn,frac,tabl,easy
%O 1,6
%A _Wolfdieter Lang_, Nov 30 2004
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