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%I #21 Mar 02 2024 13:10:24
%S 0,0,1,1,10,200,5915,229376,10946964,618435840,40283203125,
%T 2968444272640,243926836708126,22100985366992896,2187905889450121295,
%U 234881024000000000000,27172548942138551952680,3369317755618569294053376,445726953911853022186520169
%N Number of labeled 3-trees with n nodes.
%D F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30, Problem 1.13(b) with k=3.
%H T. D. Noe, <a href="/A036362/b036362.txt">Table of n, a(n) for n=1..100</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F a(n) = binomial(n, 3)*(3*n-8)^(n-5).
%F Number of labeled k-trees on n nodes is binomial(n, k) * (k(n-k)+1)^(n-k-2).
%p [ seq(binomial(n,3)*(3*n-8)^(n-5), n=1..20) ];
%t Table[Binomial[n,3](3n-8)^(n-5),{n,20}] (* _Harvey P. Dale_, Dec 31 2023 *)
%o (Python)
%o def A036362(n): return int(n*(n - 2)*(n - 1)*(3*n - 8)**(n - 5)//6) # _Chai Wah Wu_, Feb 03 2022
%Y Column 4 of A135021.
%Y Cf. A000272 (labeled trees), A036361 (labeled 2-trees), this sequence (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).
%K nonn,easy
%O 1,5
%A _N. J. A. Sloane_
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