A013525 ...and 35+ other cosec expansions?

Ralf Stephan ralf at ark.in-berlin.de
Thu Dec 16 18:11:57 CET 2004

the sequence is completely wrong IMHO:

%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 (shift), so we have:

%F A000364 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)! = 2*arctanh(cosec(x)-cotan(x)).

Now, point checks of 3 other cosec expansion show discrepancies so
I have doubts if these are all wrong. Can you confirm my suspicion?

%N A013516 exp(cosec(x)-cotan(x))=1+x+1/5!*x^5+2/6!*x^6+7/7!*x^7+23/8!*x^8...
%N A012860 log(cosec(x)*tanh(x))=-3/2!*x^2+2/4!*x^4-756/6!*x^6+272/8!*x^8...
%N A012862 arccosh(cosec(x)*arctanh(x))=x+35/3!*x^3+14/5!*x^5+389/7!*x^7...


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