A035055 Number of forests of different trees.

1, 1, 1, 2, 3, 6, 12, 24, 49, 105, 231, 517, 1188, 2783, 6643, 16101, 39606, 98605, 248287, 631214, 1618878, 4183964, 10889305, 28517954, 75111521, 198851386, 528929895, 1412993746, 3789733399, 10201625514, 27555373561, 74664487653, 202908119046, 552939614498

Offset: 0

Author

Christian G. Bower, Oct 15 1998

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A005195.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n<2, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
h:= proc(n) option remember; `if`(n=0, 1, b(n)-(add(b(k)*b(n-k),
      k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2)
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(h(i), j)*g(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> g(n, n):
seq(a(n), n=0..40); # Alois P. Heinz, May 19 2013

Mathematica

nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
b = Flatten[
  sol = SolveAlways[
    0 == Series[
      t[x] - x Product[1/(1 - x^i)^ a[i], {i, 1, nn}], {x, 0, nn}],
    x]; Table[a[n], {n, 0, nn}] /. sol];
r[x_] := Sum[b[[n]] x^(n - 1), {n, 1, nn + 1}]; c =
Drop[CoefficientList[
   Series[r[x] - (r[x]^2/2 - r[x^2]/2), {x, 0, nn}], x],
  1]; CoefficientList[
Series[Product[(1 + x^i)^c[[i]], {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Nov 15 2014 *)

Formula

Weigh transform of A000055.

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148175..., c = 0.89246007934060351292465521837... . - Vaclav Kotesovec, Aug 25 2014