%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 n1 isolated nodes and a labeled tree with n+1 nodes.
%C 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)
%F a(n) = C(2*n,n1) * (n+1)^(n1).
%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 12.
%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)^(n1) = 3 trees to complete the forest. The 12 forests can be represented by:
%e 1 324, 1 234, 1 243,
%e 2 314, 2 134, 2 143,
%e 3 214, 3 124, 3 142,
%e 4 213, 4 123, 4 132.
%t a[n_] := Binomial[2n, n1] * (n+1)^(n1); Array[a,18] (* _Amiram Eldar_, Apr 12 2020 *)
%o (PARI) a(n) = binomial(2*n,n1) * (n+1)^(n1);
%Y Cf. A000272, A001791, A302112.
%K nonn,easy
%O 1,2
%A _Washington Bomfim_, Apr 12 2020
