A309313 - OEIS

%I #13 Dec 29 2020 10:49:55

%S 1,1,19,540,23597,1381695,101682724,9016296289,935625630797,

%T 111226656560877,14903545528332565,2222230881719482634,

%U 364942065096639623872,65448490334085989020670,12726830901257817750060165,2667188536603107740647377075,599286881811684624273478547325

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

%C (a(n)/n!)^(1/n) tends to 15.1198... - _Vaclav Kotesovec_, Aug 06 2019

%H Alois P. Heinz, <a href="/A309313/b309313.txt">Table of n, a(n) for n = 0..310</a>

%F a(n) = A215861(2n,n).

%p b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,

%p       `if`(n=0, 1, add(binomial(n-1, i)*b(n-1-i, k-1)*

%p       `if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))

%p     end:

%p a:= n-> b(2*n, n):

%p seq(a(n), n=0..20);

%t b[n_, k_] := b[n, k] = If[k < 0 || k > n, 0,

%t     If[n == 0, 1, Sum[Binomial[n - 1, i]*b[n - 1 - i, k - 1]*

%t     If[i<2, 1, i!/2 + (i+1)^(i-1)], {i, 0, n-k}]]];

%t a[n_] := b[2n, n];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)

%Y Cf. A215861.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 22 2019