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A186448 E.g.f. A(x) satisfies 2*A(x) = x*(1 + A(x) + exp(A(x))).

%I A186448 #23 Oct 13 2017 05:11:44
%S A186448 1,2,10,88,1144,19856,432464,11348352,348715392,12286859008,
%T A186448 488470565632,21633197775872,1056315874429952,56382210082129920,
%U A186448 3266205054434912256,204097766901573320704,13684668496370094407680
%N A186448 E.g.f. A(x) satisfies 2*A(x) = x*(1 + A(x) + exp(A(x))).
%C A186448 A(x) is e.g.f. for a(n)/(2^(n-1)/n). - _Vaclav Kotesovec_, Jan 26 2014
%H A186448 Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A186448 a(n+1) = (n!/(2*n+2)) * (Sum_{m=1..n} binomial(n+1, m) * (Sum_{i=0..n} (m^i/i!) * binomial(n-m+1, n-i))) + (n+1)^(n-1)/2 + n!/2.
%F A186448 a(n) ~ n^(n-2) * (1+c)^(n+1) / (2 * c^n * exp(n)), where c = LambertW(exp(-1)) = 0.278464542761... (see A202357). - _Vaclav Kotesovec_, Jan 26 2014
%t A186448 Table[(Rest[CoefficientList[InverseSeries[Series[(2*x)/(1+E^x+x),{x,0,20}],x],x]*Range[0,20]!])[[n]] * (2^(n-1)/n),{n,1,20}] (* _Vaclav Kotesovec_, Jan 26 2014 *)
%t A186448 Table[n!/(2*n+2)*Sum[Binomial[n+1,m]*Sum[m^(i)/i!*Binomial[n-m+1,n-i],{i,0,n}],{m,1,n}]+(n+1)^(n-1)/2+n!/2,{n,0,20}] (* _Vaclav Kotesovec_ after _Vladimir Kruchinin_, Jan 26 2014 *)
%o A186448 (Maxima)
%o A186448 a(n):=n!/(2*n+2)*sum(binomial(n+1,m)*sum(m^(i)/i!*binomial(n-m+1,n-i),i,0,n),m,1,n)+(n+1)^(n-1)/2+(n)!/2;
%Y A186448 Cf. A202357.
%K A186448 nonn
%O A186448 1,2
%A A186448 _Vladimir Kruchinin_, Feb 22 2011