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%I #10 Aug 14 2013 15:14:44
%S 0,1,2,6,48,360,4320,60480,1008000,18869760,410054400,9859449600,
%T 262492876800,7634327500800,241483866624000,8237040844032000,
%U 301832166924288000,11812023799640064000,492028821658902528000,21728004544824754176000,1014150336304416030720000
%N Number of labeled rooted identity trees on n nodes (rooted trees that admit n! labelings).
%C a(n) = n! * A004111(n).
%H Alois P. Heinz, <a href="/A228159/b228159.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f. A(x) satisfies A(x) = x Product_{n>=1} x*(1 + x^n)^(a(n)/n!).
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(b(i-1$2), j)*b(n-i*j, i-1), j=0..n/i)))
%p end:
%p a:= n-> n!*b(n-1$2):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 14 2013
%t nn=20;SolveAlways[
%t 0 == Series[
%t f[x] - x Product[(1 + x^i)^(a[i]/i!), {i, 1, nn}], {x, 0, nn}],
%t x]; Table[a[n], {n, 0, nn}] /. sol
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Aug 14 2013
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