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
KEYWORD
nonn,frac,easy
AUTHOR
STATUS
approved