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A035055 Number of forests of different trees.

%I A035055 #23 Nov 15 2014 17:51:32
%S A035055 1,1,1,2,3,6,12,24,49,105,231,517,1188,2783,6643,16101,39606,98605,
%T A035055 248287,631214,1618878,4183964,10889305,28517954,75111521,198851386,
%U A035055 528929895,1412993746,3789733399,10201625514,27555373561,74664487653,202908119046,552939614498
%N A035055 Number of forests of different trees.
%H A035055 Alois P. Heinz, <a href="/A035055/b035055.txt">Table of n, a(n) for n = 0..1000</a>
%H A035055 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A035055 Weigh transform of A000055.
%F A035055 a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148175..., c = 0.89246007934060351292465521837... . - _Vaclav Kotesovec_, Aug 25 2014
%p A035055 with(numtheory):
%p A035055 b:= proc(n) option remember; `if`(n<2, n,
%p A035055       (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
%p A035055     end:
%p A035055 h:= proc(n) option remember; `if`(n=0, 1, b(n)-(add(b(k)*b(n-k),
%p A035055       k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2)
%p A035055     end:
%p A035055 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A035055       add(binomial(h(i), j)*g(n-i*j, i-1), j=0..n/i)))
%p A035055     end:
%p A035055 a:= n-> g(n, n):
%p A035055 seq(a(n), n=0..40); # _Alois P. Heinz_, May 19 2013
%t A035055 nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
%t A035055 b = Flatten[
%t A035055   sol = SolveAlways[
%t A035055     0 == Series[
%t A035055       t[x] - x Product[1/(1 - x^i)^ a[i], {i, 1, nn}], {x, 0, nn}],
%t A035055     x]; Table[a[n], {n, 0, nn}] /. sol];
%t A035055 r[x_] := Sum[b[[n]] x^(n - 1), {n, 1, nn + 1}]; c =
%t A035055 Drop[CoefficientList[
%t A035055    Series[r[x] - (r[x]^2/2 - r[x^2]/2), {x, 0, nn}], x],
%t A035055   1]; CoefficientList[
%t A035055 Series[Product[(1 + x^i)^c[[i]], {i, 1, nn}], {x, 0, nn}], x] (* _Geoffrey Critzer_, Nov 15 2014 *)
%Y A035055 Cf. A005195.
%K A035055 nonn
%O A035055 0,4
%A A035055 _Christian G. Bower_, Oct 15 1998