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

%I A036280 #87 Feb 23 2025 04:31:34
%S A036280 1,1,7,31,127,73,1414477,8191,16931177,5749691557,91546277357,
%T A036280 3324754717,1982765468311237,22076500342261,65053034220152267,
%U A036280 925118910976041358111,16555640865486520478399,8089941578146657681,29167285342563717499865628061
%N A036280 Numerators in Taylor series for x * cosec(x).
%C A036280 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
%D A036280 G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%H A036280 Seiichi Manyama, <a href="/A036280/b036280.txt">Table of n, a(n) for n = 0..275</a> (terms 0..100 from T. D. Noe)
%H A036280 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
%H A036280 H. W. Gould and W. Squire, <a href="http://www.jstor.org/stable/2312783">Maclaurin's second formula and its generalization</a>, Amer. Math. Monthly, 70 (1963), pp. 44-52.
%H A036280 M. Kauers and P. Paule, <a href="http://dx.doi.org/10.1007/978-3-7091-0445-3">The Concrete Tetrahedron</a>, Springer 2011, p. 30.
%H A036280 J. Malenfant, <a href="http://arxiv.org/abs/1104.4332">Factorization of and Determinant Expressions for the Hypersums of Powers of Integers</a>, arXiv preprint arXiv:1104.4332 [math.NT], 2011.
%H A036280 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HyperbolicCosecant.html">Hyperbolic Cosecant</a>.
%H A036280 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cosecant.html">Cosecant</a>.
%H A036280 Herbert S. Wilf, <a href="https://www2.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, Academic Press, NY, 1994. See p. 54.
%F A036280 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
%F A036280 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
%F A036280 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
%F A036280 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
%F A036280 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
%F A036280 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
%e A036280 cosec(x) = x^(-1) + (1/6)*x + (7/360)*x^3 + (31/15120)*x^5 + ...
%e A036280 1, 1/6, 7/360, 31/15120, 127/604800, 73/3421440, 1414477/653837184000, 8191/37362124800, ...
%p A036280 series(x*csc(x),x,60);
%p A036280 seq(numer((-1)^n*bernoulli(2*n,1/2)/(2*n)!), n=0..30); # _Robert Israel_, Mar 21 2016
%t A036280 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 *)
%o A036280 (Maxima)
%o A036280 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 */
%o A036280 (Sage)
%o A036280 def A036280_list(len):
%o A036280     R, C = [1], [1]+[0]*(len-1)
%o A036280     for n in (1..len-1):
%o A036280         for k in range(n, 0, -1):
%o A036280             C[k] = -C[k-1] / (8*k*(2*k+1))
%o A036280         C[0] = -sum(C[k] for k in (1..n))
%o A036280         R.append(C[0].numerator())
%o A036280     return R
%o A036280 print(A036280_list(19)) # _Peter Luschny_, Feb 20 2016
%o A036280 (Maxima)
%o A036280 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)!;
%o A036280 /* _Vladimir Kruchinin_, Mar 21 2016 */
%o A036280 (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
%o A036280 (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
%Y A036280 Cf. A036281, A036282, A036283.
%Y A036280 Cf. A230265.
%K A036280 nonn,frac,easy
%O A036280 0,3
%A A036280 _N. J. A. Sloane_