A332958 Number of labeled forests with 2n nodes consisting of n-1 isolated nodes and a labeled tree with n+1 nodes.

1, 12, 240, 7000, 272160, 13311144, 787218432, 54717165360, 4375800000000, 396040894180360, 40038615905992704, 4473490414613093328, 547532797546896179200, 72869747140722656250000, 10478808079059531910348800, 1619337754490833097114916960

Offset: 1

Author

Washington Bomfim, Apr 12 2020

Keywords

• nonn
• easy

Comments

Given 2n vertices, we can choose n-1 of them in C(2n, n-1) ways. For each of these ways there are A000272(n+1) trees. (possibilities)

Examples

			a(1) = 1. The forest is the tree of 2 nodes. It can be depicted by 1--2.
a(2) = 12. Given 4 nodes we can choose 1 of them in C(4,1) = 4 ways. For each of these 4 ways there are A000272(n+1) = (n+1)^(n-1) = 3 trees to complete the forest. The 12 forests can be represented by:
1  3-2-4,   1  2-3-4,   1  2-4-3,
2  3-1-4,   2  1-3-4,   2  1-4-3,
3  2-1-4,   3  1-2-4,   3  1-4-2,
4  2-1-3,   4  1-2-3,   4  1-3-2.
		

Crossrefs

Cf. A000272, A001791, A302112.

Programs

Mathematica

a[n_] := Binomial[2n, n-1] * (n+1)^(n-1); Array[a,18] (* Amiram Eldar, Apr 12 2020 *)

PARI

a(n) = binomial(2*n,n-1) * (n+1)^(n-1);

Formula

a(n) = C(2*n,n-1) * (n+1)^(n-1).

a(n) = A001791(n) * A000272(n+1).

a(n) ~ exp(1) * 2^(2*n) * n^(n - 3/2) / sqrt(Pi).