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%I M4403 N1858 #100 Jan 24 2024 09:16:24
%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(2*n, 1/2) and Bernoulli(2*n, 1/4).
%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
%C See A062715 for a method of obtaining the cosecant numbers from the square of Pascal's triangle. - _Peter Bala_, Jul 18 2013
%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 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.
%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 Masanobu Kaneko, Maneka Pallewatta, and Hirofumi Tsumura, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Tsumura/tsumura3.html">On Polycosecant Numbers</a>, J. Integer Seq. 23 (2020), no. 6, 17 pp. See line k=1 of Table 1 p. 3.
%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((-Pi^2)^(-n)*Integral_{x=0..1} (log(x/(1-x)))^2*n). - _Groux Roland_, Nov 10 2009
%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} a(n)/A001897(n)*x^(2*n)/(2*n)!. - _Peter Bala_, Jul 18 2013
%F a(n) = numerator((-1)^n*I(n)), where I(n) = 2*Pi*Integral_{z=-oo..oo} (z^n / (exp(-Pi*z) + exp(Pi*z)))^2. - _Peter Luschny_, Jul 25 2021
%e 1, -1/12, 7/240, -31/1344, 127/3840, -2555/33792, 1414477/5591040, -57337/49152, 118518239/16711680, ... = a(n)/A033469(n).
%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, ... = a(n)/A001897(n).
%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 A001896_list(18) # _Peter Luschny_, Feb 20 2016
%Y Cf. A001897 (denominators), A033469, A036280, A062715, A145901.
%Y Cf. A132092-A132099.
%K sign,frac
%O 0,3
%A _N. J. A. Sloane_
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