A036281 Denominators in Taylor series for x * cosec(x).

1, 6, 360, 15120, 604800, 3421440, 653837184000, 37362124800, 762187345920000, 2554547108585472000, 401428831349145600000, 143888775912161280000, 846912068365871834726400000, 93067260259985915904000000, 2706661834818276108533760000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Examples

			cosec(x) = x^(-1)+1/6*x+7/360*x^3+31/15120*x^5+...
1, 1/6, 7/360, 31/15120, 127/604800, 73/3421440, 1414477/653837184000, 8191/37362124800, ...
		

References

• 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).
• G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..224 (terms 0..100 from T. D. Noe)
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• 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).
• M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 30.
• J. Malenfant, Factorization of and Determinant Expressions for the Hypersums of Powers of Integers, arXiv preprint arXiv:1104.4332 [math.NT], 2011.
• Eric Weisstein's World of Mathematics, Hyperbolic Cosecant.
• Eric Weisstein's World of Mathematics, Cosecant.
• Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.
• Index entries for Bernoulli numbers B(2n).

Crossrefs

Cf. A036280, also A036282, A036283, B(2n) = A027641(2n) / A027642(2n).

Programs

Maple

series(csc(x),x,60);

Mathematica

a[n_] := 2(2^(2n-1)-1) Abs[BernoulliB[2n]]/(2n)! // Denominator;
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 14 2018 *)

Sage

def A036281_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = -C[k-1] / (k*(4*k+2))
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0].denominator())
    return R
print(A036281_list(15)) # Peter Luschny, Feb 21 2016

Formula

A036280(n)/a(n)= 2 *(2^(2n-1) -1) *abs(B(2n)) / (2n)!.

From Arkadiusz Wesolowski, Oct 16 2013: (Start)

a(n) = A036280(n)*Pi^(2*n)/(zeta(2*n)*(2 - (2^(1-n))^2)).

a(n) = A230265(n)/2. (End)