OFFSET
0,3
FORMULA
E.g.f.: A(x) = arctanh(x) o x/(1-x) o tanh(x), a composition of functions involving tanh(x) and its inverse; thus, the n-th iteration of A(x) equals arctanh(x) o x/(1-n*x) o tanh(x). [From Paul D. Hanna, Nov 29 2011]
a(n) ~ (n-1)! * 2^(n-1) / (log(3))^n. - Vaclav Kotesovec, Oct 24 2013
EXAMPLE
E.g.f.: x + 2/2!*x^2 + 6/3!*x^3 + 32/4!*x^4 + 240/5!*x^5 +...
MATHEMATICA
With[{nn=20}, CoefficientList[Series[ArcTanh[Sinh[x]Exp[x]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 14 2012 *)
PROG
(PARI){a(n)=n!*polcoeff(subst(atanh(x+x*O(x^n)), x, subst(x/(1-x), x, tanh(x+x*O(x^n)))), n)} /* Paul D. Hanna */
CROSSREFS
KEYWORD
nonn
AUTHOR
Patrick Demichel (patrick.demichel(AT)hp.com)
EXTENSIONS
Prepended a(0)=0, Joerg Arndt, Oct 14 2012
STATUS
approved