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%I #22 Aug 17 2023 08:15:45
%S 1,1,5,40,485,7776,156457,3788800,107414505,3491200000,128019454541,
%T 5229222395904,235490648957005,11592449531084800,619331166211640625,
%U 35691050995648823296,2206955604752999720273,145757527499874820423680,10240455593560436925898645
%N E.g.f. satisfies A(x) = f(x*A(x)), where f(x) = exp(x+x^2).
%C Radius of convergence of A(x): r = (1/2)*exp(-3/4) = 0.23618..., where A(r) = exp(3/4) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.
%H Seiichi Manyama, <a href="/A088695/b088695.txt">Table of n, a(n) for n = 0..364</a>
%F a(n) = n! * [x^n] exp(x+x^2)^(n+1)/(n+1).
%F a(n) = n! * Sum_{k=floor(n/2)..n} binomial(k,n-k)*(n+1)^(k-1)/k!. - _Vladimir Kruchinin_, Aug 04 2011
%F a(n) ~ 2^(n+1/2) * n^(n-1) / (sqrt(3) * exp(n/4 - 3/4)). - _Vaclav Kotesovec_, Jan 24 2014
%t Table[n!*SeriesCoefficient[(E^(x+x^2))^(n+1)/(n+1),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 24 2014 *)
%o (PARI) a(n)=n!*polcoeff(exp(x+x^2)^(n+1)+x*O(x^n),n,x)/(n+1)
%Y Cf. A143768, A362771.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 07 2003
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