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%I #13 Aug 04 2014 07:49:00
%S 1,1,3,13,80,621,5887,65689,844587,12289825,199702646,3584177829,
%T 70418168977,1503204079573,34644744039375,857391850897201,
%U 22677415997829788,638386960029846921,19057447729907765407,601346850250707128125
%N E.g.f. satisfies: A(x)+1/2*A(x)^2 = x*exp(A(x)).
%F a(n) = n!*T(n,1), T(n,m) = sum(k=1..n-m, T(n-m,k)*m^k/k!-binomial(m,k)/2^k*T(n,k+m))), n>m, with T(n,n)=1.
%F a(n) = ((n-1)!*sum(k=1..n-1, C(n+k-1,n-1)*sum(j=1..k, (-1)^(j)*C(k,j) *sum(i=0..n-1, ((-1)^i*j^i*C(j,n-i-1)*2^(-n+i+1))/i!)))), n>1, a(n)=1. - _Vladimir Kruchinin_, Feb 24 2012
%F a(n) ~ 2^(-1/4) * exp((sqrt(2)-1)*n) * (sqrt(2)-1)^(n-1/2) * n^(n-1). - _Vaclav Kotesovec_, Aug 04 2014
%t Rest[CoefficientList[InverseSeries[Series[(x*(2 + x))/(2*E^x),{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Aug 04 2014 *)
%o (Maxima)
%o array(B,100,100);
%o fillarray (B, makelist (-1, i, 1, 1000));
%o T(n,m):=if B[n,m]=-1 then BB[n,m]:(if n=m then 1 else sum(T(n-m,k)*m^k/k!-binomial(m,k)/2^k*T(n,k+m),k,1,n-m)) else B[n,m];
%o makelist(n!*T(n,1),n,1,20);
%o a(n):=if n=1 then 1 else ((n-1)!*sum(binomial(n+k-1,n-1) *sum((-1)^(j) *binomial(k,j)*sum(((-1)^i*j^i*binomial(j,n-i-1) *2^(-n+i+1))/i!, i,0,n-1), j,1,k), k,1,n-1)); [From Vladimir Kruchinin, Feb 24 2012]
%K nonn
%O 1,3
%A _Vladimir Kruchinin_, Dec 05 2011
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