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A309313 Number of simple labeled graphs on 2n nodes with exactly n connected components that are trees or cycles.

%I A309313 #13 Dec 29 2020 10:49:55
%S A309313 1,1,19,540,23597,1381695,101682724,9016296289,935625630797,
%T A309313 111226656560877,14903545528332565,2222230881719482634,
%U A309313 364942065096639623872,65448490334085989020670,12726830901257817750060165,2667188536603107740647377075,599286881811684624273478547325
%N A309313 Number of simple labeled graphs on 2n nodes with exactly n connected components that are trees or cycles.
%C A309313 (a(n)/n!)^(1/n) tends to 15.1198... - _Vaclav Kotesovec_, Aug 06 2019
%H A309313 Alois P. Heinz, <a href="/A309313/b309313.txt">Table of n, a(n) for n = 0..310</a>
%F A309313 a(n) = A215861(2n,n).
%p A309313 b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A309313       `if`(n=0, 1, add(binomial(n-1, i)*b(n-1-i, k-1)*
%p A309313       `if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
%p A309313     end:
%p A309313 a:= n-> b(2*n, n):
%p A309313 seq(a(n), n=0..20);
%t A309313 b[n_, k_] := b[n, k] = If[k < 0 || k > n, 0,
%t A309313     If[n == 0, 1, Sum[Binomial[n - 1, i]*b[n - 1 - i, k - 1]*
%t A309313     If[i<2, 1, i!/2 + (i+1)^(i-1)], {i, 0, n-k}]]];
%t A309313 a[n_] := b[2n, n];
%t A309313 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A309313 Cf. A215861.
%K A309313 nonn
%O A309313 0,3
%A A309313 _Alois P. Heinz_, Jul 22 2019