|
%I #45 Mar 01 2020 04:28:25
%S 1,6,360,15120,604800,3421440,653837184000,37362124800,
%T 762187345920000,2554547108585472000,401428831349145600000,
%U 143888775912161280000,846912068365871834726400000,93067260259985915904000000,2706661834818276108533760000000
%N Denominators in Taylor series for x * cosec(x).
%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).
%D G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%H Seiichi Manyama, <a href="/A036281/b036281.txt">Table of n, a(n) for n = 0..224</a> (terms 0..100 from T. D. Noe)
%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 M. Kauers and P. Paule, <a href="http://dx.doi.org/10.1007/978-3-7091-0445-3">The Concrete Tetrahedron</a>, Springer 2011, p. 30.
%H J. Malenfant, <a href="http://arxiv.org/abs/1104.4332">Factorization of and Determinant Expressions for the Hypersums of Powers of Integers</a>, arXiv preprint arXiv:1104.4332 [math.NT], 2011.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicCosecant.html">Hyperbolic Cosecant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cosecant.html">Cosecant</a>
%H Index entries for <a href="/index/Be#Bernoulli">Bernoulli numbers</a> B(2n)
%F A036280(n)/a(n)= 2 *(2^(2n-1) -1) *abs(B(2n)) / (2n)!.
%F From _Arkadiusz Wesolowski_, Oct 16 2013: (Start)
%F a(n) = A036280(n)*Pi^(2*n)/(zeta(2*n)*(2 - (2^(1-n))^2)).
%F a(n) = A230265(n)/2. (End)
%e cosec(x) = x^(-1)+1/6*x+7/360*x^3+31/15120*x^5+...
%e 1, 1/6, 7/360, 31/15120, 127/604800, 73/3421440, 1414477/653837184000, 8191/37362124800, ...
%p series(csc(x),x,60);
%t a[n_] := 2(2^(2n-1)-1) Abs[BernoulliB[2n]]/(2n)! // Denominator;
%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jul 14 2018 *)
%o (Sage)
%o def A036281_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] / (k*(4*k+2))
%o C[0] = -sum(C[k] for k in (1..n))
%o R.append(C[0].denominator())
%o return R
%o print(A036281_list(15)) # _Peter Luschny_, Feb 21 2016
%Y Cf. A036280, also A036282, A036283, B(2n) = A027641(2n) / A027642(2n).
%K nonn,frac,easy
%O 0,2
%A _N. J. A. Sloane_
| 