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%I #71 Aug 12 2019 04:06:11
%S 1,7,31,127,511,1414477,8191,118518239,5749691557,91546277357,
%T 162912981133,1982765468311237,22076500342261,455371239541065869,
%U 925118910976041358111,16555640865486520478399,1302480594081611886641,904185845619475242495834469891
%N Write cosec x = 1/x + Sum_{n>=1} e_n * x^(2n-1)/(2n-1)!; sequence gives numerators of e_n.
%C From _Johannes W. Meijer_, May 24 2009: (Start)
%C Absolute value of numerator of [2^(2n-1) - 1] * Bernoulli(2n)/n.
%C Equals the absolute values of the numerators of the LS1[ -2*m,n=1] matrix coefficients of A160487 for m = 1, 2, .. ,.
%C (End)
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
%H Seiichi Manyama, <a href="/A036282/b036282.txt">Table of n, a(n) for n = 1..275</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
%H R. P. Brent, <a href="http://arxiv.org/abs/1608.04834"> Asymptotic approximation of central binomial coefficients with rigorous error bounds</a>, arXiv:1608.04834 [math.NA], 2016.
%H Duane W. DeTemple, Shun-Hwa Wang, <a href="http://dx.doi.org/10.1016/0022-247X(91)90296-C">Half-integer approximations for the partial sums of harmonic series</a>, J. Math. Anal. Applic. 160 (1991) 149-156
%H Simon Plouffe, <a href="http://arxiv.org/abs/1310.7195">On the values of the functions zeta and gamma</a>, arXiv:1310.7195 [math.NT], 2013.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cosecant.html">Cosecant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Riemann-SiegelFunctions.html">Riemann-Siegel Function</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Trigonometric_functions">Trigonometric functions</a>
%e cosec x
%e = x^(-1) + 1/6*x + 7/360*x^3 + 31/15120*x^5 + ...
%e = x^(-1) + 1/6 * x/1! + 7/60 * x^3/3! + 31/126 * x^5/5! + ...
%p a:= n-> (m-> numer(coeff(series(csc(x), x, m+1), x, m)*m!))(2*n-1):
%p seq(a(n), n=1..20); # _Alois P. Heinz_, Jun 21 2018
%t a[n_] := Abs[ Numerator[ (2^(2*n-1)-1) * BernoulliB[2*n]/n ] ]; Table[a[n], {n, 1, 18}] (* _Jean-François Alcover_, May 31 2013, after _Johannes W. Meijer_ *)
%o (PARI) a(n) = abs(numerator((2^(2*n-1)-1)*bernfrac(2*n)/n)); \\ _Michel Marcus_, Mar 01 2015
%Y Cf. A036280, A036281, A036283.
%Y Cf. A160487.
%Y Differs from A282898.
%K nonn,frac,easy
%O 1,2
%A _N. J. A. Sloane_
%E Title corrected and offset changed by _Johannes W. Meijer_, May 21 2009
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