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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)
22

%I M4403 N1858

%S 1,-1,7,-31,127,-2555,1414477,-57337,118518239,-5749691557,

%T 91546277357,-1792042792463,1982765468311237,-286994504449393,

%U 3187598676787461083,-4625594554880206790555,16555640865486520478399,-22142170099387402072897

%N Numerators of cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)}; also of Bernoulli(2n,1/2) and Bernoulli(2n,1/4).

%C Cosecant number are given by the integral: (-Pi^2)^(-n)*int((log(x/(1-x)))^2*n,x=0..1) - _Groux Roland_, Nov 10 2009

%C |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

%D 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.

%D S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51.

%D N. E. Nörlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 458.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Hector Blandin and Rafael Diaz, <a href="http://arXiv.org/abs/0708.0809">Compositional Bernoulli numbers</a>, arXiv:0708.0809 [math.CO], 2007-2008; Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897.

%H S. A. Joffe, <a href="/A001896/a001896.pdf">Sums of like powers of natural numbers</a>, Quart. J. Pure Appl. Math. 46 (1914), 33-51. [Annotated scanned copy of pages 38-51 only, plus notes]

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., 36 (1935), 637-649.

%H N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%F a(n) = numerator((-1)^(n+1)*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(-1)). - _Peter Luschny_, Jun 29 2012

%F 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} A001896(n)/A001897(n)*x^(2*n)/(2*n)!. - _Peter Bala_, Jul 18 2013

%F See A062715 for a method of obtaining the cosecant numbers from the square of Pascal's triangle. - _Peter Bala_, Jul 18 2013

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

%e 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.

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

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

%o (PARI) a(n) = numerator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ _Michel Marcus_, Mar 01 2015

%o (Sage)

%o def A001896_list(len):

%o R, C = [1], [1]+[0]*(len-1)

%o for n in (1..len-1):

%o for k in range(n, 0, -1):

%o C[k] = C[k-1] / (8*k*(2*k+1))

%o C[0] = -sum(C[k] for k in (1..n))

%o R.append((C[0]*factorial(2*n)).numerator())

%o return R

%o print A001896_list(18) # _Peter Luschny_, Feb 20 2016

%Y Cf. A001897, A033469, A036280, A132092...A132106, A062715, A145901.

%K sign,frac

%O 0,3

%A _N. J. A. Sloane_.

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