A036280 Numerators in Taylor series for x * cosec(x).

1, 1, 7, 31, 127, 73, 1414477, 8191, 16931177, 5749691557, 91546277357, 3324754717, 1982765468311237, 22076500342261, 65053034220152267, 925118910976041358111, 16555640865486520478399, 8089941578146657681, 29167285342563717499865628061

Offset

0,3

Comments

These are also the numerators of the coefficients appearing in the MacLaurin summation formula (which might be called the 'MacLaurin numbers') (see Gould & Squire, p. 45). - Peter Luschny, Feb 20 2016

References

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..275 (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, p. 75 (4.3.68).

H. W. Gould and W. Squire, Maclaurin's second formula and its generalization, Amer. Math. Monthly, 70 (1963), pp. 44-52.

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

Formula

Numerator of Sum_{k=1..2*n-2} Sum_{j=1..k} 2^(1-j)*(-1)^(n+j-1) * binomial(k,j) * Sum_{i=0..floor(j/2)} (j-2*i)^(2*n+j-2) * binomial(j,i) * (-1)^i/(2*n+j-2)!, n > 1. - Vladimir Kruchinin, Apr 12 2011

E.g.f.: x/sin(x) = 1 + (x^2/(6-x^2))*T(0), where T(k) = 1 - x^2*(2*k+2)*(2*k+3)/( x^2*(2*k+2)*(2*k+3) + ((2*k+2)*(2*k+3) - x^2)*((2*k+4)*(2*k+5) - x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013

a(n) = numerator((-1)^n*B(2*n,1/2)/(2*n)!) where B(n,x) denotes the Bernoulli polynomial. - Peter Luschny, Feb 20 2016

a(n) = numerator(Sum_{k=1..n+1}((Sum_{j=2*k-1..2*n+1}(binomial(j,2*k-1)*(j-1)!*2^(1-j)*(-1)^(n+1+j)*stirling2(2*n+1,j)))/(2*k-1))/(2*n)!). - Vladimir Kruchinin, Mar 21 2016

a(n) = numerator(eta(2*n)/Pi^(2*n)), where eta(n) is the Dirichlet eta function. See A230265 for denominator. - Mohammed Yaseen, Aug 02 2023

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(x*csc(x), x, 60);
seq(numer((-1)^n*bernoulli(2*n, 1/2)/(2*n)!), n=0..30); # Robert Israel, Mar 21 2016

Mathematica

nn = 34; t = Numerator[CoefficientList[Series[x*Csc[x], {x, 0, nn}], x]*Range[0, nn]!]; Take[t, {1, nn-1, 2}] (* T. D. Noe, Oct 28 2013 *)

Prog

(Maxima)
a(n):=num(sum(sum((2^(1-j)*(-1)^(n+j-1)*binomial(k, j)*sum((j-2*i)^(2*n+j-2)*binomial(j, i)*(-1)^(i), i, 0, floor(j/2)))/(2*n+j-2)!, j, 1, k), k, 1, 2*n-2)); n>1. a(1)=1. /* Vladimir Kruchinin, Apr 12 2011 */
(Sage)
def A036280_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] / (8*k*(2*k+1))
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0].numerator())
    return R
print(A036280_list(19)) # Peter Luschny, Feb 20 2016
(Maxima)
a(n):=(sum((sum(binomial(j, 2*k-1)*(j-1)!*2^(1-j)*(-1)^(n+1+j)*stirling2(2*n+1, j), j, 2*k-1, 2*n+1))/(2*k-1), k, 1, n+1))/(2*n)!;
/* Vladimir Kruchinin, Mar 21 2016 */

Crossrefs

Cf. A036281, A036282, A036283.

Cf. A230265.

Sequence in context: A002184 A002588 A327497 * A153005 A056909 A002147

Adjacent sequences: A036277 A036278 A036279 * A036281 A036282 A036283

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved