A227465 E.g.f. equals the series reversion of arctan(x) / exp(x).

1, 2, 11, 96, 1141, 17232, 316175, 6831104, 169889641, 4780648960, 150175445331, 5209500696576, 197793228285277, 8158536901294080, 363292669599123287, 17369586234209861632, 887496174440659597009, 48261023190850955378688, 2782898587468279374050715

Offset: 1

Author

Paul D. Hanna, Jul 14 2013

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 96*x^4/4! + 1141*x^5/5! + 17232*x^6/6! + ...
where A( arctan(x)/exp(x) ) = x.
		

Crossrefs

Cf. A227466.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[ArcTan[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(atan(X)/exp(X)), n)}
for(n=1,25,print1(a(n),", "))

PARI

{a(n)=local(A=x); for(i=1,n,A=tan(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) = tan(x*exp(A(x))).

a(n) ~ n^(n-1) * ((1+s^2)/exp(1-s))^n * sqrt(1+s^2)/(1+s), where s = 0.74721195516156756882... is the root of the equation (1+s^2)*arctan(s) = 1. - Vaclav Kotesovec, Jan 13 2014