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%I #10 Aug 04 2014 06:57:06
%S 1,1,3,15,109,1053,12767,186763,3204313,63128665,1404963387,
%T 34867190823,954800951749,28600649870133,930325531322519,
%U 32658109219519843,1230609634110271921,49545182501048868145
%N E.g.f. exp(A006351(x)).
%F a(n) = sum(m=1..n, (sum(k=0..n-m, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(l=0..j, (2^(j-l)*(-1)^(l+j)*stirling1(n-m-l+j,j-l))/(l!*(n-m-l+j)!)))))/(m-1)!), n>0, a(0)=1.
%F E.g.f.: 4*LambertW(-exp((x-1)/2)/2)^2 / exp(x). - _Vaclav Kotesovec_, Aug 04 2014
%F a(n) ~ sqrt(2) * n^(n-1) / (exp(n-1) * (2*log(2)-1)^(n-1/2)). - _Vaclav Kotesovec_, Aug 04 2014
%t CoefficientList[Series[4*ProductLog[-E^((x-1)/2)/2]^2/E^x,{x, 0, 15}], x]*Range[0, 15]! (* _Vaclav Kotesovec_, Aug 04 2014 *)
%o (Maxima)
%o a(n):=(sum((m*sum((n+k-1)!*sum(1/(k-j)!*sum((2^(j-l)*(-1)^(l+j)*stirling1(n-m-l+j,j-l))/(l!*(n-m-l+j)!),l,0,j),j,0,k),k,0,n-m))/m!,m,1,n));
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Sep 26 2012
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