A006965 Number of directed trees with n nodes.

1, 2, 6, 25, 114, 591, 3298, 19532, 120687, 771373, 5061741, 33943662, 231751331, 1606587482, 11283944502, 80157645245, 575105238243, 4162624144308, 30365913761136, 223075674659696, 1649166676341180, 12262121068089094, 91649977839972636, 688288656744067230

Offset: 1

Author

Simon Plouffe

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1093
• P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Quebec, 1992, Vol. 16, No. 1, 53-80.
• P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
• Index entries for sequences related to trees

Crossrefs

Equals (1/3) A038060(n).

Row sums of A335362.

Programs

Maple

with(combstruct):B:=x->add(3*count([S, {B = Set(S), S = Prod(B,B,B,Z)}, unlabeled],size=i)*x^i,i=1..50);seq(coeff(B(x)-B(x)^2/2+B(x^2)/2,x,n)/3,n=1..30);# with Algolib (Pab Ter)
# second Maple program:
b:= proc(n) option remember; `if`(n<2, 3*n, (add(add(b(d)
      *d, d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= n-> `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)/3:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 03 2020

Mathematica

b[n_] := b[n] = If[n < 2, 3 n, (Sum[Sum[b[d] d, {d, Divisors[j]}] b[n - j], {j, 1, n - 1}])/(n - 1)];
a[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]/3;
Array[a, 30] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)