

A332958


Number of labeled forests with 2n nodes consisting of n1 isolated nodes and a labeled tree with n+1 nodes.


1



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

1,2


COMMENTS

Given 2n vertices, we can choose n1 of them in C(2n, n1) ways. For each of these ways there are A000272(n+1) trees. (possibilities)


LINKS



FORMULA

a(n) = C(2*n,n1) * (n+1)^(n1).
a(n) ~ exp(1) * 2^(2*n) * n^(n  3/2) / sqrt(Pi).


EXAMPLE

a(1) = 1. The forest is the tree of 2 nodes. It can be depicted by 12.
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)^(n1) = 3 trees to complete the forest. The 12 forests can be represented by:
1 324, 1 234, 1 243,
2 314, 2 134, 2 143,
3 214, 3 124, 3 142,
4 213, 4 123, 4 132.


MATHEMATICA

a[n_] := Binomial[2n, n1] * (n+1)^(n1); Array[a, 18] (* Amiram Eldar, Apr 12 2020 *)


PROG

(PARI) a(n) = binomial(2*n, n1) * (n+1)^(n1);


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



