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A358732 Number of labeled trees covering 2n nodes, half of which are leaves.

%I A358732 #13 Aug 02 2024 11:36:48
%S A358732 0,12,720,109200,31752000,15186346560,10852244282880,
%T A358732 10851787634688000,14481281691676800000,24881574582258352358400,
%U A358732 53525038934303849706393600,140958354488116955062668595200,446153762528143389466306560000000,1671353230826683972965623004979200000
%N A358732 Number of labeled trees covering 2n nodes, half of which are leaves.
%H A358732 Andrew Howroyd, <a href="/A358732/b358732.txt">Table of n, a(n) for n = 1..100</a>
%F A358732 a(n) = A055314(2*n, n) = Stirling2(2*n-2, n)*(2*n)!/n!. - _Andrew Howroyd_, Dec 30 2022
%e A358732 The a(2) = 12 trees:
%e A358732   {{1,2},{1,3},{2,4}}
%e A358732   {{1,2},{1,3},{3,4}}
%e A358732   {{1,2},{1,4},{2,3}}
%e A358732   {{1,2},{1,4},{3,4}}
%e A358732   {{1,2},{2,3},{3,4}}
%e A358732   {{1,2},{2,4},{3,4}}
%e A358732   {{1,3},{1,4},{2,3}}
%e A358732   {{1,3},{1,4},{2,4}}
%e A358732   {{1,3},{2,3},{2,4}}
%e A358732   {{1,3},{2,4},{3,4}}
%e A358732   {{1,4},{2,3},{2,4}}
%e A358732   {{1,4},{2,3},{3,4}}
%t A358732 a[n_]:=StirlingS2[2*n-2, n]*(2*n)!/n!; Array[a,14] (* _Stefano Spezia_, Aug 02 2024 *)
%o A358732 (PARI) a(n) = stirling(2*n-2, n, 2)*(2*n)!/n! \\ _Andrew Howroyd_, Dec 30 2022
%Y A358732 A central column of A055314.
%Y A358732 The unlabeled rooted version is A185650.
%Y A358732 The unlabeled version is A358107.
%Y A358732 A000272 counts trees, bisection A163395.
%Y A358732 A001187 counts connected graphs.
%Y A358732 A006129 counts covering graphs.
%Y A358732 A014068 counts graphs with n vertices and n-1 edges.
%Y A358732 Cf. A000055, A001349, A006125.
%K A358732 nonn
%O A358732 1,2
%A A358732 _Gus Wiseman_, Dec 01 2022
%E A358732 Terms a(6) and beyond from _Andrew Howroyd_, Dec 30 2022