A194770 E.g.f. 2*sqrt(3)/3*arctan(sqrt(3)*x/(x+2)).

1, -1, 0, 6, -24, 0, 720, -5040, 0, 362880, -3628800, 0, 479001600, -6227020800, 0, 1307674368000, -20922789888000, 0, 6402373705728000, -121645100408832000, 0, 51090942171709440000, -1124000727777607680000

Offset: 1

Author

Peter Bala, Sep 02 2011

Keywords

• sign
• easy

Crossrefs

A080635

Programs

Mathematica

With[{nn=30},Rest[CoefficientList[Series[2 Sqrt[3]/3 ArcTan[Sqrt[ 3] x/(x+2)],{x,0,nn}],x] Range[0,nn-1]!]] (* Harvey P. Dale, May 13 2019 *)

Formula

a(3*n+1) = (3*n)!, a(3*n+2) = -(3*n+1)!, a(3*n) = 0.

E.g.f.: A(x) = 2*sqrt(3)/3*arctan(sqrt(3)*x/(x+2)) = x-x^2/2!+6*x^4/4!-24*x^5/5!+720*x^7/7!-....

The derivative A'(x) = 1/(1+x+x^2). The inverse function A^-1(x) = 2/sqrt(3)*tan(sqrt(3)/2*x)/(1-1/sqrt(3)*tan(sqrt(3)/2*x)) is the generating function for A080635 (apart from the initial term).

D-finite with recurrence: a(n) +(n-1)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Jan 25 2020