A036280 - OEIS

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

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

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).

Eric Weisstein's World of Mathematics, Cosecant.

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

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, ...

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

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 */

(Sage)

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

(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)!;