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A001897
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Denominators of cosecant numbers -2*(2^(2*n-1)-1)*Bernoulli(2*n).
(Formerly M2983 N1205)
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8
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1, 3, 15, 21, 15, 33, 1365, 3, 255, 399, 165, 69, 1365, 3, 435, 7161, 255, 3, 959595, 3, 6765, 903, 345, 141, 23205, 33, 795, 399, 435, 177, 28393365, 3, 255, 32361, 15, 2343, 70050435, 3, 15, 1659, 115005, 249, 1702155, 3, 30705, 136059, 705, 3, 2250885, 3, 16665, 2163
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OFFSET
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0,2
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REFERENCES
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H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 187.
S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51.
N. E. Nörlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 458.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=0..51.
Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer 1924, p. 27.
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FORMULA
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a(0)=1, a(n)=1/2*A002445(n) for n>=1. [Joerg Arndt, May 07 2012]
a(n) = denominator((2*n)!*Li_{2*n}(1)) for n > 0. - Peter Luschny, Jun 29 2012
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EXAMPLE
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Cosecant numbers -2*(2^(2*n-1)-1)*Bernoulli(2*n) are 1, -1/3, 7/15, -31/21, 127/15, -2555/33, 1414477/1365, -57337/3, 118518239/255, -5749691557/399, 91546277357/165, -1792042792463/69, 1982765468311237/1365, -286994504449393/3, 3187598676787461083/435, ... = A001896/A001897.
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MAPLE
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b := n -> bernoulli(n)*2^add(i, i=convert(n, base, 2));
a := n -> denom(b(2*n)); # Peter Luschny, May 02 2009
# Alternative :
Clausen := proc(n) local i, S; map(i->i+1, numtheory[divisors](n));
S := select(isprime, %); if S <> {} then mul(i, i=S) else NULL fi end:
A001897_list := n -> [1, seq(Clausen(2*i)/2, i=1..n-1)];
A001897_list(52); # Peter Luschny, Oct 03 2011
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CROSSREFS
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Cf. A001896, A132092-A132106, A160014.
Sequence in context: A083545 A097571 A048087 * A074214 A036897 A129966
Adjacent sequences: A001894 A001895 A001896 * A001898 A001899 A001900
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KEYWORD
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nonn,frac
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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