A038057 - OEIS

%I #26 Mar 14 2023 09:31:20

%S 2,8,72,1024,20000,497664,15059072,536870912,22039921152,

%T 1024000000000,53119845582848,3043362286338048,190857913323364352,

%U 13004222844995895296,956593800000000000000,75557863725914323419136

%N a(n) = 2^n*n^(n-1).

%C Labeled rooted trees with n 2-colored nodes.

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F E.g.f.: B(2*x) where B(x) is e.g.f. of A000169.

%F Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e. - _Stefano Spezia_, Mar 12 2023

%t nn=16;f[x_]:=Sum[a[n]x^n/n!,{n,0,nn}];s=SolveAlways[0==Series[f[x]-2x Exp[f[x]],{x,0,nn}],x];Table[a[n],{n,1,nn}]/.s

%t (* or *)

%t nn=16;Drop[Range[0,nn]!CoefficientList[Series[-LambertW[-2x],{x,0,nn}],x],1]

%t (* or *)

%t Table[2^n*n^(n-1),{n,1,16}]  (* _Geoffrey Critzer_, Mar 17 2013 *)

%Y Cf. A000169, A019762 (2*e), A038055-A038062.

%Y Equals 2 * A052746(n).

%K nonn

%O 1,1

%A _Christian G. Bower_, Jan 04 1999

%E New description from _Vladeta Jovovic_, Mar 08 2003