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A001896 Numerators of cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)}; also of Bernoulli(2n,1/2) and Bernoulli(2n,1/4).
(Formerly M4403 N1858)
9
1, -1, 7, -31, 127, -2555, 1414477, -57337, 118518239, -5749691557, 91546277357, -1792042792463, 1982765468311237, -286994504449393, 3187598676787461083, -4625594554880206790555, 16555640865486520478399, -22142170099387402072897 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Cosecant number are given by the integral: (-Pi^2)^(-n)*int((ln(x/(1-x)))^2*n,x=0..1) [From Groux Roland, Nov 10 2009]

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.

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-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..17.

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897.

Index entries for sequences related to Bernoulli numbers.

FORMULA

a(n) = numerator((-1)^(n+1)*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(-1)). - Peter Luschny, Jun 29 2012

E.g.f. 2*x*exp(x)/(exp(2*x) - 1) = 1 - 1/3*x^2/2! + 7/15*x^4/4! - 31/21*x^6/6! + .... = sum {n >= 0} A0001896(n)/A001897(n)*x^(2*n)/(2*n)!. - Peter Bala, Jul 18 2013

See A062715 for a method of obtaining the cosecant numbers from the square of Pascal's triangle. - Peter Bala, Jul 18 2013

EXAMPLE

1, -1/12, 7/240, -31/1344, 127/3840, -2555/33792, 1414477/5591040, -57337/49152, 118518239/16711680, ... = A001896/A033469

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

with(numtheory); [ seq(numer(bernoulli(2*n, 1/2)), n=0..20) ];

MATHEMATICA

a[n_] := -2*(2^(2*n-1)-1)*BernoulliB[2*n]; Table[a[n], {n, 0, 20}] // Numerator (* Jean-François Alcover, Sep 11 2013 *)

CROSSREFS

Cf. A001897, A033469, A132092-A132106. A062715, A145901.

Sequence in context: A083420 A036282 A033474 * A180147 A044049 A005825

Adjacent sequences:  A001893 A001894 A001895 * A001897 A001898 A001899

KEYWORD

sign,frac

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 19 14:18 EDT 2014. Contains 240760 sequences.