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%I #10 Aug 30 2019 03:33:33
%S 1,1,1,1,4,2,1,9,6,2,1,16,24,32,8,1,25,200,40,40,8,1,36,150,160,360,
%T 32,16,1,49,294,490,280,56,112,16,1,64,1568,6272,1120,3584,1792,1024,
%U 128,1,81,864,14112,18144,2016,5376,6912,1152,128,1,100,1350,5760,10080,8064
%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), 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="/A101020/a101020.txt">Rational polynomials R(n,x)</a>
%F a(n, m)= numerator(R(n, x)[x^m]), m=0, ..., n, n=0, 1, ..., with the rational polynomials R(n, x) of degree n defined by R(n, x):=hypergeom([ -n, -n], [1/2], x/2) = 1 + sum(r(n, m)*x^m, m=1..n), n>=0.
%F The rational polynomials are R(n, x) = 1 + sum(((binomial(n, m)^2)/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)^2)/binomial(2*m, m)), m=1..n, n=1, 2, ... and r(n, 0)=1, n>=0, else 0.
%e The rows of the rational table are: [1/1]; [1/1,1/1]; [1/1,4/1,2/3]; [1/1, 9/1, 6/1, 2/5]; ...
%Y The denominator table is given in A101021.
%K nonn,frac,tabl,easy
%O 0,5
%A _Wolfdieter Lang_, Nov 30 2004
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