A036361 Number of labeled 2-trees with n nodes.

0, 1, 1, 6, 70, 1215, 27951, 799708, 27337500, 1086190605, 49162945645, 2496308717826, 140489907594114, 8678436279296875, 583701359488329915, 42457773984656284920, 3320786296452525792376, 277898747312921495246937, 24775177557380767822265625

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30.

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• L. W. Beineke and R. E. Pipert, The number of labeled k-dimensional trees, J. Comb. Theory 6 (2) (1969) 200-205. Math. Rev. 38 #3182.
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• T. Fowler, I. Gessel, G. Labelle and P. Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, eq. (18).
• Index entries for sequences related to trees

Crossrefs

Column 3 of A135021.

Cf. A000272 (labeled trees), this sequence (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).

Programs

Maple

A036361:=n->binomial(n, 2)*(2*n-3)^(n-4): seq(A036361(n), n=1..30);

Mathematica

Table[Binomial[n,2](2n-3)^(n-4),{n,20}] (* Harvey P. Dale, Nov 24 2011 *)

Python

def A036361(n): return int(n*(n - 1)*(2*n - 3)**(n - 4)//2) # Chai Wah Wu, Feb 03 2022

Formula

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