OFFSET
1,2
COMMENTS
Radius of convergence is r = 0.22053155301... = A(r) - (1-A(r)^2)^2/4, where A(r) = tanh( (1-A(r)^2)/2 ) = 0.39770222711593...
FORMULA
E.g.f. satisfies:
(1) A(x) = Series_Reversion( x - arctanh(x)^2 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) arctanh(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (arctanh(x)^(2*n)/x) / n! ).
(4) A'(x) = (1 - A(x)^2) / (1 - A(x)^2 - 2*arctanh(A(x))).
a(n) ~ ((1-s^2)/sqrt(2+4*s*arctanh(s))) * n^(n-1) / (exp(n) * r^(n-1/2)), where r and A(r) were described above, s = A(r) is the root of the equation 2*arctanh(s) = 1-s^2 and r = s-arctanh(s)^2. - Vaclav Kotesovec, Jan 12 2014
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 136*x^4/4! + 2160*x^5/5! + ...
Related expansions:
arctanh(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 160*x^4/4! + 2544*x^5/5! + ...
arctanh(A(x))^2 = 2*x^2/2! + 12*x^3/3! + 136*x^4/4! + 2160*x^5/5! + ...
Series expressions:
A(x) = x + arctanh(x)^2 + d/dx arctanh(x)^4/2! + d^2/dx^2 arctanh(x)^6/3! + d^3/dx^3 arctanh(x)^8/4! + ...
log(A(x)/x) = arctanh(x)^2/x + d/dx (arctanh(x)^4/x)/2! + d^2/dx^2 (arctanh(x)^6/x)/3! + d^3/dx^3 (arctanh(x)^8/x)/4! + ...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x - ArcTanh[x]^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *)
PROG
(PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=x + atanh(A)^2); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(serreverse(x-atanh(x+x*O(x^n))^2), n)}
for(n=1, 18, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, atanh(x+x*O(x^n))^(2*m)/m!)); n!*polcoeff(A, n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, atanh(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 03 2012
STATUS
approved