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A101026
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Table of numerators of coefficients of certain rational polynomials.
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2
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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.
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REFERENCES
| H. E. Haber and H. A. Weldon, On the relativistic Bose-Einstein integrals, J. Math. Phys. 23(10) (1982) 1852-1858.
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LINKS
| W. Lang: Rational polynomials R(n,x)
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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.
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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];...
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CROSSREFS
| The table of denominators is given in A101027.
Sequence in context: A065451 A178148 A054506 * A055249 A125172 A073732
Adjacent sequences: A101023 A101024 A101025 * A101027 A101028 A101029
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KEYWORD
| nonn,frac,tabl,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Nov 30 2004
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