cp's OEIS Frontend

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

%I A332958 #35 May 06 2020 09:41:35
%S A332958 1,12,240,7000,272160,13311144,787218432,54717165360,4375800000000,
%T A332958 396040894180360,40038615905992704,4473490414613093328,
%U A332958 547532797546896179200,72869747140722656250000,10478808079059531910348800,1619337754490833097114916960
%N A332958 Number of labeled forests with 2n nodes consisting of n-1 isolated nodes and a labeled tree with n+1 nodes.
%C A332958 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 A332958 a(n) = C(2*n,n-1) * (n+1)^(n-1).
%F A332958 a(n) = A001791(n) * A000272(n+1).
%F A332958 a(n) ~ exp(1) * 2^(2*n) * n^(n - 3/2) / sqrt(Pi).
%e A332958 a(1) = 1. The forest is the tree of 2 nodes. It can be depicted by 1--2.
%e A332958 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 A332958 1  3-2-4,   1  2-3-4,   1  2-4-3,
%e A332958 2  3-1-4,   2  1-3-4,   2  1-4-3,
%e A332958 3  2-1-4,   3  1-2-4,   3  1-4-2,
%e A332958 4  2-1-3,   4  1-2-3,   4  1-3-2.
%t A332958 a[n_] := Binomial[2n, n-1] * (n+1)^(n-1); Array[a,18] (* _Amiram Eldar_, Apr 12 2020 *)
%o A332958 (PARI) a(n) = binomial(2*n,n-1) * (n+1)^(n-1);
%Y A332958 Cf. A000272, A001791, A302112.
%K A332958 nonn,easy
%O A332958 1,2
%A A332958 _Washington Bomfim_, Apr 12 2020