f[n_,x_] = If[ n==1, 1/(1-x), 1/(1-(3^(n-1)-2^n+2)/3^(n-1)x) x/3^(n-1) Sum[Binomial[n,m]f[m,x],{m,1,n-1}]]
Series[f[2,x],{x,0,10}]
Series[f[3,x],{x,0,10}]
Series[f[4,x],{x,0,10}]
Series[f[11,x],{x,0,10}]