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A006790 Exponentiation of e.g.f. for trees A000055(n-1).

%I A006790 #16 Jul 08 2025 00:49:29
%S A006790 1,2,5,15,53,211,938,4582,24349,139671,858745,5628789,39145021,
%T A006790 287667582,2226033629,18082308403,153770703339,1365631349757,
%U A006790 12638233544989,121640399661294,1215438543434225,12587691428792115
%N A006790 Exponentiation of e.g.f. for trees A000055(n-1).
%H A006790 Alois P. Heinz, <a href="/A006790/b006790.txt">Table of n, a(n) for n = 0..500</a>
%H A006790 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%p A006790 with(numtheory):
%p A006790 b:= proc(n) option remember; `if`(n<=1, n, add(add(d*
%p A006790       b(d), d=divisors(j))*b(n-j), j=1..n-1)/(n-1))
%p A006790     end:
%p A006790 t:= proc(n) option remember; `if`(n=0, 1, b(n)-(add(b(k)
%p A006790       *b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
%p A006790     end:
%p A006790 g:= proc(n) option remember; `if`(n=0, 1, add(
%p A006790       binomial(n-1, j-1) *t(j-1) *g(n-j), j=1..n))
%p A006790     end:
%p A006790 a:= n-> g(n+1):
%p A006790 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 16 2015
%t A006790 b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]; t[n_] := t[n] = If[n==0, 1, b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[ Mod[n, 2] == 0, b[n/2], 0])/2]; g[n_] := g[n] = If[n==0, 1, Sum[Binomial[n-1, j-1] *t[j-1]*g[n-j], {j, 1, n}]]; a[n_] := g[n+1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 30 2015, after _Alois P. Heinz_ *)
%Y A006790 Cf. A000055.
%K A006790 nonn
%O A006790 0,2
%A A006790 _N. J. A. Sloane_