A101026 Table of numerators of coefficients of certain rational polynomials.

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, 240, 128, 64, 1, 63, 504, 1176, 1008, 336, 448, 192, 1, 80, 840, 2688, 3360, 1792, 4480, 5120, 128, 1, 99, 1320, 5544, 9504, 7392, 2688, 5760, 384, 128, 1, 120, 1980, 10560

Offset

0,3

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), l. 4, l=2 case (space dimension 3), 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.

Links

Table of n, a(n) for n=0..58.

W. Lang: Rational polynomials R(n,x)

Formula

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.

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.

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.

Example

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];...

Crossrefs

The table of denominators is given in A101027.

Sequence in context: A337681 A245180 A054506 * A055249 A125172 A344381

Adjacent sequences: A101023 A101024 A101025 * A101027 A101028 A101029

Keyword

nonn,frac,tabl,easy

Author

Wolfdieter Lang, Nov 30 2004

Status

approved