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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

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

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

• 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.
• Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

Crossrefs

Cf. A036281, A036282, A036283.

Cf. A230265.

Programs

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 *)

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 */

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 */

PARI

a(n)=numerator(sum(k=1,n,sum(j=0,k/2,binomial(3*n,n-k)*(-1)^(n+j)*(2*j-k)^(2*n+k)*2^(n+1-k)*(n+1)!/(j!*(k-j)!*(k+1))))/((3*n)!*2^n))+(n==0) \\ Tani Akinari, Feb 22 2025

PARI

my(x='x+O('x^40), v=apply(numerator, Vec(x/sin(x)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Feb 23 2025

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

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

a(n) = numerator((Sum_{k=1..n} Sum_{j=0..floor(k/2)} binomial(3*n,n-k)*(-1)^(n+j)*(2*j-k)^(2*n+k)*2^(n+1-k)*(n+1)!/(j!*(k-j)!*(k+1)))/((3*n)!*2^n)) for n > 0. - Tani Akinari, Feb 22 2025