A038052 Number of labeled trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

1, 1, 2, 7, 42, 376, 4513, 68090, 1238968, 26416729, 646140364, 17837852044, 548713088399, 18612963873492, 690271321314292, 27785827303491579, 1206582732097720126, 56224025231569020724, 2798445211000659147033, 148178324442139816854902, 8317074395027724691495980

Offset: 0

Author

Christian G. Bower, Jan 04 1999

Keywords

• nonn
• nice
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..379 (first 101 terms from T. D. Noe)
• Index entries for sequences related to trees

Crossrefs

Cf. A036250, A048802.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0,
      m^max(0, m-2), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 16 2022

Mathematica

a[0] = 1; a[n_] := Sum[StirlingS2[n, k]*k^(k - 2), {k, 1, n}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Sep 09 2013, after Vladeta Jovovic *)

Formula

E.g.f.: B(e^x-1) where B is e.g.f. of A000272.

a(n) = Sum_{k=1..n} Stirling2(n, k)*k^(k-2). - Vladeta Jovovic, Sep 20 2003

a(n) ~ (1+exp(1))^(3/2) * n^(n-2) / (exp(n) * (log(1+exp(-1)))^(n-3/2)). - Vaclav Kotesovec, Feb 17 2017