OFFSET
1,2
COMMENTS
Note that arcsinh(x) = log(sqrt(1+x^2) + x).
FORMULA
E.g.f. A(x) satisfies:
(1) A(x - x*arcsinh(x)) = x.
(2) A(x) = x/(1 - arcsinh(A(x))).
(3) A(x) = sinh( (A(x) - x) / A(x) ).
(4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * arcsinh(x)^n / n!.
(5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * arcsinh(x)^n / n! ).
a(n) ~ n^(n-1) * s*sqrt((1+s^2)/(2+s^2)) / (exp(n) * (s^2/sqrt(1+s^2))^n), where s = 0.54523200963327675622... is the root of the equation s = sqrt(1+s^2)*(1-arcsinh(s)). - Vaclav Kotesovec, Jan 13 2014
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 116*x^4/4! + 1560*x^5/5! + ...
where A(x) = x/(1 - arcsinh(A(x))).
The e.g.f. satisfies:
(4) A(x) = x + x*arcsinh(x) + d/dx x^2*arcsinh(x)^2/2! + d^2/dx^2 x^3*arcsinh(x)^3/3! + d^3/dx^3 x^4*arcsinh(x)^4/4! + ...
(5) log(A(x)/x) = arcsinh(x) + d/dx x*arcsinh(x)^2/2! + d^2/dx^2 x^2*arcsinh(x)^3/3! + d^3/dx^3 x^3*arcsinh(x)^4/4! + ...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x - x*ArcSinh[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)
PROG
(PARI) {a(n)=n!*polcoeff(serreverse(x-x*asinh(x +x*O(x^n))), n)}
for(n=1, 25, 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, x^m*asinh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(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, x^(m-1)*asinh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 13 2013
STATUS
approved