A215853 Number of simple labeled graphs on n nodes with exactly 3 connected components that are trees or cycles.

1, 6, 55, 540, 6412, 90734, 1515097, 29368155, 649910349, 16178495157, 447436384356, 13607804913248, 451277483034618, 16204761730619392, 626327433705523558, 25924177756443661632, 1144012780063556028591, 53615833082093775740400, 2659498185704802765924159

Offset: 3

Author

Alois P. Heinz, Aug 25 2012

Keywords

• nonn

Examples

			a(4) = 6:
.1-2.  .1 2.  .1 2.  .1 2.  .1 2.  .1 2.
.   .  .  |.  .   .  .|  .  . \ .  . / .
.4 3.  .4 3.  .4-3.  .4 3.  .4 3.  .4 3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..145

Crossrefs

Column k=3 of A215861.

The unlabeled version is A215983.

Programs

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, add(binomial(n-1, i)*T(n-1-i, k-1)*
      `if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
    end:
a:= n-> T(n, 3):
seq(a(n), n=3..25);

Mathematica

T[n_, k_] := T[n, k] = If[k<0 || k>n, 0, If[n == 0, 1, Sum[Binomial[n-1, i]*T[n-1-i, k-1]*If[i<2, 1, i!/2 + (i+1)^(i-1)], {i, 0, n-k}]]];
a[n_] := T[n, 3];
Table[a[n], {n, 3, 25}] (* Jean-François Alcover, Apr 01 2017, translated from Maple *)

Formula

a(n) ~ c * n^(n-2), where c = 0.130848879059... . - Vaclav Kotesovec, Sep 07 2014