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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

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

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 *)

Crossrefs

Cf. A005195.

Sequence in context: A042950 A098011 A367222 * A119559 A375578 A329675

Adjacent sequences: A035052 A035053 A035054 * A035056 A035057 A035058

Keyword

nonn

Author

Christian G. Bower, Oct 15 1998

Status

approved