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A101022
Table of numerators of coefficients of certain rational polynomials.
2
1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 4, 2, 8, 1, 5, 4, 1, 8, 4, 1, 1, 2, 2, 8, 8, 16, 1, 7, 14, 1, 8, 4, 16, 2, 1, 4, 56, 4, 16, 32, 64, 16, 128, 1, 3, 8, 6, 16, 8, 64, 8, 128, 64, 1, 5, 2, 12, 16, 16, 160, 16, 128, 128, 256, 1, 11, 22, 33, 176, 8, 32, 4, 128, 64, 256, 64, 1, 2, 44, 22, 88, 32
OFFSET
1,6
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. 2, 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-1, n>=1, with the rational polynomials R(n, x) of degree n-1 defined by R(n, x):=hypergeom([1, 1, 1-n], [3/2, 2], -x/2)) = sum(R(n, m)*x^m, m=0..n-1), n>=1.
The rational polynomials are R(n, x) = 1 + sum(binomial(n-1, m)/((m+1)*(2*m+1)*binomial(2*m, m))*(2*x)^m, m=1..n-1), n>=1.
a(n, m)=numerator(R(n, m)) with R(n, m) = binomial(n-1, m)/((m+1)*(2*m+1)*binomial(2*m, m))*2^m, m=1..n-1, n=1, 2, ... and R(n, 0)=1, n>=1, else 0.
EXAMPLE
The rows of the rational table are: [1/1]; [1/1, 1/6]; [1/1, 1/3, 2/45]; [1/1, 1/2, 2/15, 1/70]; ...
CROSSREFS
The table of denominators is given in A101023.
Sequence in context: A136610 A326371 A226304 * A241153 A213852 A051064
KEYWORD
nonn,frac,tabl,easy
AUTHOR
Wolfdieter Lang, Nov 30 2004
STATUS
approved