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A088690 E.g.f.: A(x) = f(x*A(x)), where f(x) = (1+x)*exp(x).

%I A088690 #15 Apr 20 2016 12:38:12
%S A088690 1,2,11,106,1489,27696,643579,17973488,586899009,21953140480,
%T A088690 925890264331,43480125312768,2250352192663249,127280062346049536,
%U A088690 7811329076598534075,517016126622623635456,36713034605774835974401,2784127167066690618458112
%N A088690 E.g.f.: A(x) = f(x*A(x)), where f(x) = (1+x)*exp(x).
%C A088690 Radius of convergence of A(x): r = tau^2*exp(-tau) = 0.20588... and A(r) = (1+tau)*exp(tau), where tau=(sqrt(5)-1)/2 and r = limit a(n)/a(n+1)*n as n->infinity.
%H A088690 Alois P. Heinz, <a href="/A088690/b088690.txt">Table of n, a(n) for n = 0..200</a>
%F A088690 a(n) = n! * [x^n] ((1+x)*exp(x))^(n+1)/(n+1).
%F A088690 a(n) = Sum_{k=1..n} n^(k-2)*n!/k!*binomial(n-1,k-1) (offset 1). - _Vladeta Jovovic_, Jun 17 2006
%F A088690 E.g.f.: A(x) = (1/x)*series_reversion(x*exp(-x)/(1+x)). - _Paul D. Hanna_, Jun 17 2006
%F A088690 E.g.f.: B(x)/(1-x*B(x)), where B(x) is e.g.f. for A052873(). - _Vladeta Jovovic_, Jun 18 2006
%F A088690 a(n) ~ 5^(-1/4) * ((1+sqrt(5))/2)^(2*n+2) * exp((sqrt(5) - 1 - (3 - sqrt(5))*n)/2) * n^(n-1). - _Vaclav Kotesovec_, Jan 24 2014
%F A088690 a(n) = n!*hypergeom([-n], [2], -n-1). - _Peter Luschny_, Apr 20 2016
%p A088690 a := n -> n!*simplify(hypergeom([-n], [2], -n-1)):
%p A088690 seq(a(n), n=0..15); # _Peter Luschny_, Apr 20 2016
%t A088690 CoefficientList[1/x*InverseSeries[Series[x*E^(-x)/(1+x), {x, 0, 21}], x],x]*Range[0, 20]! (* _Vaclav Kotesovec_, Jan 24 2014 *)
%o A088690 (PARI) a(n)=n!*polcoeff(((1+x)*exp(x))^(n+1)+x*O(x^n),n,x)/(n+1)
%K A088690 nonn
%O A088690 0,2
%A A088690 _Paul D. Hanna_, Oct 06 2003