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%I #35 May 06 2020 09:41:35
%S 1,12,240,7000,272160,13311144,787218432,54717165360,4375800000000,
%T 396040894180360,40038615905992704,4473490414613093328,
%U 547532797546896179200,72869747140722656250000,10478808079059531910348800,1619337754490833097114916960
%N Number of labeled forests with 2n nodes consisting of n-1 isolated nodes and a labeled tree with n+1 nodes.
%C 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)
%F a(n) = C(2*n,n-1) * (n+1)^(n-1).
%F a(n) = A001791(n) * A000272(n+1).
%F a(n) ~ exp(1) * 2^(2*n) * n^(n - 3/2) / sqrt(Pi).
%e a(1) = 1. The forest is the tree of 2 nodes. It can be depicted by 1--2.
%e 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:
%e 1 3-2-4, 1 2-3-4, 1 2-4-3,
%e 2 3-1-4, 2 1-3-4, 2 1-4-3,
%e 3 2-1-4, 3 1-2-4, 3 1-4-2,
%e 4 2-1-3, 4 1-2-3, 4 1-3-2.
%t a[n_] := Binomial[2n, n-1] * (n+1)^(n-1); Array[a,18] (* _Amiram Eldar_, Apr 12 2020 *)
%o (PARI) a(n) = binomial(2*n,n-1) * (n+1)^(n-1);
%Y Cf. A000272, A001791, A302112.
%K nonn,easy
%O 1,2
%A _Washington Bomfim_, Apr 12 2020
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