A013518 Numerator of [x^(2n+1)] in the Taylor expansion arcsin(cosec(x)-cot(x)) = x/2 + x^3/16 + 3*x^5/256 + 83*x^7/30720 + 8887*x^9/12386304 + ...

1, 1, 3, 83, 8887, 57539, 2419601, 298733192941, 84896691713, 54207578317691, 535009143553922969, 303988210353762448529, 39439620915967757710853, 18146112662693896499335287481

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• nonn
• frac

Comments

The e.g.f. of x/2, arcsin(cosec(x)-cot(x)) = x/(2^1*1!) + 3*x^3/(2^3*3!) + 45*x^5/(2^5*5!) +1743*x^7(/2^7*7!) + 133305*x^9/(2^9*9!) + ..., is apparently covered by A012780.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..216

Crossrefs

Cf. A089171.

Programs

Mathematica

Numerator[Take[CoefficientList[Series[ArcSin[Csc[x]-Cot[x]],{x,0,30}], x],{2,-1,2}]] (* Harvey P. Dale, Feb 02 2012 *)

Maxima

a(n):=(sum((binomial(2*k,k)*sum(binomial(j+2*k,2*k)*(j+2*k+1)!*2^(-4*k-j-1)*(-1)^(n+k+j)*stirling2(2*n+1,j+2*k+1),j,0,2*n-2*k))/(2*k+1),k,0,n))/(2*n+1)!; /* Vladimir Kruchinin, May 31 2013 */

Formula

a(n)=(sum(k=0..n, (binomial(2*k,k)*sum(j=0..2*n-2*k, binomial(j+2*k,2*k)*(j+2*k+1)!*2^(-4*k-j-1)*(-1)^(n+k+j)*stirling2(2*n+1,j+2*k+1)))/(2*k+1)))/(2*n+1)!. - Vladimir Kruchinin, May 31 2013

Extensions

Name edited by R. J. Mathar, Dec 19 2011