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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) - Groux Roland, Nov 10 2009

|A001896(n)|*pi^(2n)/A001897(n) is the value of the multi zeta function z(2,2,...,2) with n 2's, where z(k_l,k_2,...,k_n) = sum_{i_n >= i_(n-1) >= ... >= i_1 >= 1}1/((i_1)^k_1 (i_2)^k_2 ... (i_n)^k_n). The proof is simple: start with the product expansion sin(pi x)/(pi x) = product_{r>=1}(1-x^2/r^2), take reciprocals, and expand the right side. The coefficient of x^(2n) is seen to be z(2,2,...,2) with n 2's. - David Callan, Aug 27 2014

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..17.

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.

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

[ 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 December 22 10:11 EST 2014. Contains 252339 sequences.