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

0,2

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

Index entries for Bernoulli numbers B(2n)

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)

Example

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, ...

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A290782 A367519 A002684 * A202367 A262179 A064350

Adjacent sequences: A036278 A036279 A036280 * A036282 A036283 A036284

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved