A101020 Table of numerators of coefficients of certain rational polynomials.

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.

Links

Table of n, a(n) for n=0..60.

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

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

Adjacent sequences: A101017 A101018 A101019 * A101021 A101022 A101023

Keyword

nonn,frac,tabl,easy

Author

Wolfdieter Lang, Nov 30 2004

Status

approved