A036362 Number of labeled 3-trees with n nodes.

0, 0, 1, 1, 10, 200, 5915, 229376, 10946964, 618435840, 40283203125, 2968444272640, 243926836708126, 22100985366992896, 2187905889450121295, 234881024000000000000, 27172548942138551952680, 3369317755618569294053376, 445726953911853022186520169

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30, Problem 1.13(b) with k=3.

Links

• T. D. Noe, Table of n, a(n) for n=1..100
• Index entries for sequences related to trees

Crossrefs

Column 4 of A135021.

Cf. A000272 (labeled trees), A036361 (labeled 2-trees), this sequence (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).

Programs

Maple

[ seq(binomial(n,3)*(3*n-8)^(n-5), n=1..20) ];

Mathematica

Table[Binomial[n,3](3n-8)^(n-5),{n,20}] (* Harvey P. Dale, Dec 31 2023 *)

Python

def A036362(n): return int(n*(n - 2)*(n - 1)*(3*n - 8)**(n - 5)//6) # Chai Wah Wu, Feb 03 2022

Formula

a(n) = binomial(n, 3)*(3*n-8)^(n-5).

Number of labeled k-trees on n nodes is binomial(n, k) * (k(n-k)+1)^(n-k-2).