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

%I #14 Mar 30 2015 06:44:33

%S 1,2,5,15,53,211,938,4582,24349,139671,858745,5628789,39145021,

%T 287667582,2226033629,18082308403,153770703339,1365631349757,

%U 12638233544989,121640399661294,1215438543434225,12587691428792115

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

%H Alois P. Heinz, <a href="/A006790/b006790.txt">Table of n, a(n) for n = 0..500</a>

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

%p with(numtheory):

%p b:= proc(n) option remember; `if`(n<=1, n, add(add(d*

%p b(d), d=divisors(j))*b(n-j), j=1..n-1)/(n-1))

%p end:

%p t:= proc(n) option remember; `if`(n=0, 1, b(n)-(add(b(k)

%p *b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)

%p end:

%p g:= proc(n) option remember; `if`(n=0, 1, add(

%p binomial(n-1, j-1) *t(j-1) *g(n-j), j=1..n))

%p end:

%p a:= n-> g(n+1):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 16 2015

%t 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 Cf. A000055.

%K nonn

%O 0,2

%A _N. J. A. Sloane_.