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%I #10 Aug 30 2019 03:35:26
%S 1,1,3,1,8,4,1,15,20,4,1,24,60,32,24,1,35,140,140,40,8,1,48,280,448,
%T 240,128,64,1,63,504,1176,1008,336,448,192,1,80,840,2688,3360,1792,
%U 4480,5120,128,1,99,1320,5544,9504,7392,2688,5760,384,128,1,120,1980,10560
%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. 4, l=2 case (space dimension 3), 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="/A101026/a101026.txt">Rational polynomials R(n,x)</a>
%F a(n, m)= numerator(R(n, x)[x^m]), m=0, ..., n, n>=0, with the rational polynomials R(n, x) of degree n defined by R(n, x):=hypergeom([ -n, -2-n], [1/2], -x/2)) = sum(r(n, m)*x^m, m=0..n), n>=0.
%F The rational polynomials are R(n, x) = 1 + sum((binomial(n, m)*binomial(n+2, m)/binomial(2*m, m))*(2*x)^m, m=1..n), n>=0.
%F a(n, m)=numerator(r(n, m)) with the rational triangle r(n, m) = (2^m)*binomial(n, m)*binomial(n+2, m)/binomial(2*m, m), m=1..n, n>=1 and r(n, 0)=1, n>=0, else 0.
%e The rows of the rational table are: [1/1]; [1/1, 3/1]; [1/1, 8/1, 4/1]; [1/1, 15/1, 20/1, 4/1]; [1/1, 24/1, 60/1, 32/1, 24/7];...
%Y The table of denominators is given in A101027.
%K nonn,frac,tabl,easy
%O 0,3
%A _Wolfdieter Lang_, Nov 30 2004
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