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%I #26 Jun 02 2024 11:08:42
%S 1,4,30,344,5370,106452,2562182,72592816,2367054450,87320153900,
%T 3595646533182,163492924997448,8136172620013802,439858024910227588,
%U 25670670464821310070,1608575860476990991712,107716675117341985862370
%N Number of labeled rooted trees with n nodes and 2-colored internal (non-leaf) nodes.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Divides by 2n and shifts left under exponential transform.
%F E.g.f.: -x-LambertW(-2*x*exp(-x)). - _Vladeta Jovovic_, Sep 17 2003
%F a(n) = sum(j=1..n, j^(n-1)*2^j*(-1)^(n-j)*binomial(n,j)), n>1, a(1)=1. - _Vladimir Kruchinin_, Jan 24 2012
%F a(n) ~ sqrt(1+LambertW(-exp(-1)/2)) * n^(n-1) / (exp(n)*(-LambertW(-exp(-1)/2))^n). - _Vaclav Kotesovec_, Oct 05 2013
%t a[n_] := Sum[j^(n-1)*2^j*(-1)^(n-j)*Binomial[n, j], {j, 1, n}]; a[1] = 1; Table[a[n], {n, 1, 17}] (* _Jean-François Alcover_, Feb 26 2013, after _Vladimir Kruchinin_ *)
%o (Maxima) a(n):=if n=1 then 1 else sum(j^(n-1)*2^j*(-1)^(n-j)*binomial(n,j),j,1,n); /* _Vladimir Kruchinin_, Jan 24 2012 */
%Y Cf. A004113, A004114, A052317.
%K nonn,eigen
%O 1,2
%A _Christian G. Bower_, Dec 15 1999