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A001897 Denominators of cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)}.
(Formerly M2983 N1205)
12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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).

LINKS

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.

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

a[n_] := -2*(2^(2*n-1)-1)*BernoulliB[2*n]; Table[a[n], {n, 0, 51}] // Denominator (* Jean-François Alcover, Sep 11 2013 *)

CROSSREFS

Cf. A001896, A132092-A132106, A160014.

Sequence in context: A224872 A097571 A048087 * A074214 A036897 A129966

Adjacent sequences:  A001894 A001895 A001896 * A001898 A001899 A001900

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 22 03:25 EDT 2014. Contains 244801 sequences.