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

1, 1, 19, 540, 23597, 1381695, 101682724, 9016296289, 935625630797, 111226656560877, 14903545528332565, 2222230881719482634, 364942065096639623872, 65448490334085989020670, 12726830901257817750060165, 2667188536603107740647377075, 599286881811684624273478547325

Offset: 0

Author

Alois P. Heinz, Jul 22 2019

Keywords

• nonn

Comments

(a(n)/n!)^(1/n) tends to 15.1198... - Vaclav Kotesovec, Aug 06 2019

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..310

Crossrefs

Cf. A215861.

Programs

Maple

b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, add(binomial(n-1, i)*b(n-1-i, k-1)*
      `if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..20);

Mathematica

b[n_, k_] := b[n, k] = If[k < 0 || k > n, 0,
    If[n == 0, 1, Sum[Binomial[n - 1, i]*b[n - 1 - i, k - 1]*
    If[i<2, 1, i!/2 + (i+1)^(i-1)], {i, 0, n-k}]]];
a[n_] := b[2n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

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