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%I A035054 #15 Feb 19 2016 05:40:12
%S A035054 1,1,2,2,4,4,9,12,27,49,111,236,562,1302,3172,7746,19347,48630,123923,
%T A035054 317956,823178,2144518,5623993,14828075,39300482,104636894,279794753,
%U A035054 751065509,2023446206,5469566586,14830879661,40330829031,109972429568,300628862717
%N A035054 Number of forests of identical trees.
%H A035054 Alois P. Heinz, <a href="/A035054/b035054.txt">Table of n, a(n) for n = 0..750</a>
%H A035054 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A035054 Inverse Moebius transform of A000055.
%F A035054 a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148..., c = A086308 = 0.53494960614230701455... . - _Vaclav Kotesovec_, Aug 25 2014
%p A035054 with(numtheory):
%p A035054 b:= proc(n) option remember; `if`(n<=1, n,
%p A035054       (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
%p A035054     end:
%p A035054 g:= proc(n) option remember; local k; `if`(n=0, 1, b(n)-
%p A035054       (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2)
%p A035054     end:
%p A035054 a:= n-> `if`(n=0, 1, add(g(d), d=divisors(n))):
%p A035054 seq(a(n), n=0..35);  # _Alois P. Heinz_, May 18 2013
%t A035054 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)]; g[n_] := g[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]; a[n_] := If[n==0, 1, Sum[ g[d], {d, Divisors[n]}]]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Feb 19 2016, after _Alois P. Heinz_ *)
%Y A035054 Cf. A005195.
%K A035054 nonn
%O A035054 0,3
%A A035054 _Christian G. Bower_, Oct 15 1998.

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