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Search: id:A101022
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| A101022 |
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Table of numerators of coefficients of certain rational polynomials. |
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2
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| 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
(list; table; graph; listen)
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OFFSET
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1,6
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COMMENT
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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.
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REFERENCES
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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
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W. Lang: Rational polynomials R(n,x)
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FORMULA
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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.
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EXAMPLE
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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]; ...
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CROSSREFS
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The table of denominators is given in A101023.
Adjacent sequences: A101019 A101020 A101021 this_sequence A101023 A101024 A101025
Sequence in context: A029316 A104368 A012257 this_sequence A051064 A078770 A072038
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KEYWORD
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nonn,frac,tabl,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Nov 30 2004
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