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 A227465 E.g.f. equals the series reversion of arctan(x) / exp(x). 3

%I

%S 1,2,11,96,1141,17232,316175,6831104,169889641,4780648960,

%T 150175445331,5209500696576,197793228285277,8158536901294080,

%U 363292669599123287,17369586234209861632,887496174440659597009,48261023190850955378688,2782898587468279374050715

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

%F E.g.f. A(x) satisfies: A(x) = tan(x*exp(A(x))).

%F 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

%e 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! + ...

%e where A( arctan(x)/exp(x) ) = x.

%t Rest[CoefficientList[InverseSeries[Series[ArcTan[x] / Exp[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 13 2014 *)

%o (PARI) {a(n)=local(X=x+x*O(x^n));n!*polcoeff(serreverse(atan(X)/exp(X)), n)}

%o for(n=1,25,print1(a(n),", "))

%o (PARI) {a(n)=local(A=x); for(i=1,n,A=tan(x*exp(A+x*O(x^n)))); n!*polcoeff(A, n)}

%o for(n=1,25,print1(a(n),", "))

%Y Cf. A227466.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 14 2013

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Last modified December 2 05:01 EST 2021. Contains 349437 sequences. (Running on oeis4.)