A227463 E.g.f. equals the series reversion of arcsinh(x) / exp(x).

1, 2, 10, 80, 876, 12192, 206144, 4104704, 94092112, 2440642560, 70676191840, 2260198354944, 79113937385536, 3008546200346624, 123513154739070976, 5444598073252904960, 256489070938397360384, 12859678961654923395072, 683701585124386481758720

Offset: 1

Author

Paul D. Hanna, Jul 13 2013

Keywords

• nonn

Comments

Note that arcsinh(x) = log(sqrt(1+x^2) + x).

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 80*x^4/4! + 876*x^5/5! + 12192*x^6/6! + ...
where A( arcsinh(x)/exp(x) ) = x.
		

Crossrefs

Cf. A227464.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[ArcSinh[x] / Exp[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

PARI

{a(n)=local(X=x+x*O(x^n));n!*polcoeff(serreverse(asinh(X)/exp(X)), n)}
for(n=1,25,print1(a(n),", "))

PARI

{a(n)=local(A=x); for(i=1,n,A=sinh(x*exp(A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: A(x) = sinh(x*exp(A(x))).

a(n) ~ n^(n-1) * sqrt((1+s^2)/(1+s+s^2)) * (sqrt(1+s^2)/exp(1-s))^n, where s = 0.84184323411403778647... is the root of the equation sqrt(1+s^2)*arcsinh(s) = 1. - Vaclav Kotesovec, Jan 13 2014