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A101024 Table of numerators of coefficients of certain rational polynomials. 2

%I

%S 1,1,2,1,6,2,1,12,12,8,1,20,40,16,8,1,30,100,80,120,16,1,42,210,280,

%T 120,16,16,1,56,392,784,560,448,448,128,1,72,672,9408,2016,896,1792,

%U 1536,128,1,90,1080,4032,6048,4032,13440,23040,1152,256,1,110,1650,7920

%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=1 case, 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="/A101024/a101024.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, -1-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+1, 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+1, 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]; [1/1,2/1]; [1/1, 6/1, 2/1];[1/1, 12/1, 12/1, 8/5]; ...

%Y The table of denominators is given in A101025.

%K nonn,frac,tabl,easy

%O 0,3

%A _Wolfdieter Lang_, Nov 30 2004

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