A013525 ...and 35+ other cosec expansions?

Don Reble djr at nk.ca
Fri Dec 17 03:59:50 CET 2004

> %N A013525 arctanh(cosec(x)-cotan(x))=x+2/5!*x^5+30/7!*x^7+692/9!*x^9...
> please check with Mathematica; here both Pari and Maxima said
> that this is 1/2*x+1/2*x^3/3!+5/2*x^5/5!+61/2*x^7/7!+...
> and equals 1/2*sec(x) after differentiation

A trig identity might help: csc x - cot x = tan (x/2).
One differentiates atanh(tan(x/2)) easily: it is indeed (sec x)/2.
The integral of sec is inverse-gudermannian, and CRC reports
    gdinv x = sum{n=0 to inf} (-1)^n E_{2n} x^{2n+1} / (2n+1)!
So just divide the Euler numbers by +-2.
(Mathematica? What's that? :-)

(I'm using the alternating-sign version of the Euler numbers. That's why
(-1)^n appears above.)

Also, the Euler numbers are odd; so to make A013525 an _integer_
sequence, we subtracted
    sum{n=0 to inf} (1/2) x^{2n+1} / (2n+1)! = (sinh x)/2.
The sequence ends up being (gdinv x - sinh x)/2.

Don Reble  djr at nk.ca

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