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A101020
Table of numerators of coefficients of certain rational polynomials.
2
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, 32, 16, 1, 49, 294, 490, 280, 56, 112, 16, 1, 64, 1568, 6272, 1120, 3584, 1792, 1024, 128, 1, 81, 864, 14112, 18144, 2016, 5376, 6912, 1152, 128, 1, 100, 1350, 5760, 10080, 8064
OFFSET
0,5
COMMENTS
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.
REFERENCES
H. E. Haber and H. A. Weldon, On the relativistic Bose-Einstein integrals, J. Math. Phys. 23(10) (1982) 1852-1858.
FORMULA
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.
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.
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.
EXAMPLE
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]; ...
CROSSREFS
The denominator table is given in A101021.
Sequence in context: A021241 A016691 A177347 * A160905 A208612 A183157
KEYWORD
nonn,frac,tabl,easy
AUTHOR
Wolfdieter Lang, Nov 30 2004
STATUS
approved