A035056 Number of asymmetric forests with n nodes.

1, 1, 0, 0, 0, 0, 0, 1, 2, 4, 9, 21, 44, 96, 206, 450, 981, 2159, 4757, 10571, 23563, 52835, 118939, 269047, 610878, 1392677, 3186001, 7313882, 16842202, 38900699, 90098260, 209229601, 487077685, 1136549747, 2657859059, 6228447488, 14624515804, 34402798404

Offset: 0

Author

Christian G. Bower, Oct 15 1998

Keywords

• nonn

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 301 and 562.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms
• Index entries for sequences related to forests

Crossrefs

Cf. A000220, A004111, A005195, A246169.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(b((i-1)$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
g:= n-> b((n-1)$2):
h:= proc(n) option remember; g(n)-add(g(i)*g(n-i), i=0..n)/2
       -`if`(irem(n, 2)=1, 0, g(n/2))/2
    end:
f:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(h(i), j)*f(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> f(n, n):
seq(a(n), n=0..40);  # Alois P. Heinz, May 20 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[b[i-1, i-1], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; g[n_] := b[n-1, n-1]; h[n_] := h[n] = g[n] - Sum[g[i]*g[n-i], {i, 0, n}]/2 - If[Mod[n, 2]==1, 0, g[n/2]]/2; f[n_, i_] := f[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[h[i], j]*f[n - i*j, i-1], {j, 0, n/i}]]]; a[n_] := f[n, n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz *)

Formula

Weigh transform of A000220.

a(n) ~ c * d^n / n^(5/2), where d = A246169 = 2.51754035263200389079535..., c = 0.421943694962576031011358... . - Vaclav Kotesovec, Aug 25 2014