A101024 Table of numerators of coefficients of certain rational polynomials.

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, 120, 16, 16, 1, 56, 392, 784, 560, 448, 448, 128, 1, 72, 672, 9408, 2016, 896, 1792, 1536, 128, 1, 90, 1080, 4032, 6048, 4032, 13440, 23040, 1152, 256, 1, 110, 1650, 7920

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

Crossrefs

The table of denominators is given in A101025.

Sequence in context: A208751 A133200 A103881 * A124730 A114283 A106187

Adjacent sequences: A101021 A101022 A101023 * A101025 A101026 A101027

Keyword

nonn,frac,tabl,easy

Author

Wolfdieter Lang, Nov 30 2004

Status

approved