A035054 Number of forests of identical trees.

1, 1, 2, 2, 4, 4, 9, 12, 27, 49, 111, 236, 562, 1302, 3172, 7746, 19347, 48630, 123923, 317956, 823178, 2144518, 5623993, 14828075, 39300482, 104636894, 279794753, 751065509, 2023446206, 5469566586, 14830879661, 40330829031, 109972429568, 300628862717

Offset: 0

Author

Christian G. Bower, Oct 15 1998

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..750
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A005195.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
g:= proc(n) option remember; local k; `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:
a:= n-> `if`(n=0, 1, add(g(d), d=divisors(n))):
seq(a(n), n=0..35);  # Alois P. Heinz, May 18 2013

Mathematica

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 *)

Formula

Inverse Moebius transform of A000055.

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148..., c = A086308 = 0.53494960614230701455... . - Vaclav Kotesovec, Aug 25 2014