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A088692 E.g.f: A(x) = f(x*A(x)), where f(x) = (1+2*x)*exp(x).

%I A088692 #19 Oct 09 2017 00:01:24
%S A088692 1,3,23,304,5829,147696,4670371,177383424,7874174601,400298556160,
%T A088692 22940919680271,1463679309053952,102911522568495757,
%U A088692 7906731860604186624,659108356837269579675,59252790438687592677376,5714517052927568389576209,588555892122678050845556736
%N A088692 E.g.f: A(x) = f(x*A(x)), where f(x) = (1+2*x)*exp(x).
%C A088692 Radius of convergence of A(x): r = (1/4)*exp(-1/2) = 0.151632.., where A(r) = 2*exp(1/2) and r = lim_{n->infinity} (a(n)/a(n+1))*n.
%H A088692 Robert Israel, <a href="/A088692/b088692.txt">Table of n, a(n) for n = 0..342</a>
%F A088692 a(n) = n! * [x^n] ((1+2*x)*exp(x))^(n+1)/(n+1).
%F A088692 a(n) = Sum_{k=1..n} 2^(n-k)*n^(k-2)*n!/k!*binomial(n-1,k-1) (offset 1). - _Vladeta Jovovic_, Jun 19 2006
%F A088692 a(n) ~ 4^(n+1) * n^(n-1) / (sqrt(3) * exp(n/2-1/2)). - _Vaclav Kotesovec_, Jan 24 2014
%F A088692 a(n) = exp(-(n+1)/4)*2^n*n!*(n+1)^(-2)*((5*n+1)*WhittakerM(-n,1/2,(n+1)/2) - 2*(n-1)*WhittakerM(1-n,1/2,(n+1)/2)). - _Robert Israel_, Oct 08 2017
%p A088692 f := n -> simplify(exp(-(1/4)*n-1/4)*2^n*factorial(n)*((5*n+1)*WhittakerM(-n, 1/2, (1/2)*n+1/2)-(2*n-2)*WhittakerM(1-n, 1/2, (1/2)*n+1/2))/(n+1)^2):
%p A088692 map(f, [$0..30]); # _Robert Israel_, Oct 08 2017
%t A088692 Table[Sum[2^(n-k)*n^(k-2)*n!/k!*Binomial[n-1,k-1],{k,1,n}],{n,1,21}] (* _Vaclav Kotesovec_ after _Vladeta Jovovic_, Jan 24 2014 *)
%o A088692 (PARI) a(n)=n!*polcoeff(((1+2*x)*exp(x))^(n+1)+x*O(x^n),n,x)/(n+1)
%Y A088692 Cf. A088690, A088693.
%K A088692 nonn
%O A088692 0,2
%A A088692 _Paul D. Hanna_, Oct 07 2003