A108246 Number of labeled 2-regular graphs with no multiple edges, but loops are allowed (i.e., each vertex is endpoint of two (usual) edges or one loop).

1, 1, 1, 2, 8, 38, 208, 1348, 10126, 86174, 819134, 8604404, 98981944, 1237575268, 16710431992, 242337783032, 3756693451772, 61991635990652, 1084943597643964, 20072853005524696, 391443701509660096, 8024999955144721256, 172544980412641191776

Offset: 0

Author

Marni Mishna, Jun 17 2005

Keywords

• nonn

Examples

			a(3) = 2: {(1,2) (2,3) (1,3)}, {(1,1) (2,2) (3,3)}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000985, A002137.

Binomial transform of A001205.

Row sums of A144161. - Alois P. Heinz, Jun 01 2009

Programs

Maple

b:= proc(n) option remember; if n=0 then 1 elif n<3 then 0 else (n-1) *(b(n-1) +b(n-3) *(n-2)/2) fi end: a:= proc(n) add(b(k) *binomial(n,k), k=0..n) end: seq(a(n), n=0..30);  # Alois P. Heinz, Sep 12 2008

Mathematica

CoefficientList[Series[E^(-x^2/4+x/2)/Sqrt[1-x], {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 17 2012 *)

Formula

Linear recurrence satisfied by a(n): {a(2) = 1, a(0) = 1, (-n^2 - 3*n - 2)*a(n) + (4 + 2*n)*a(n+1) + (-2*n-6)*a(n+2) + 2*a(n+3), a(1) = 1}.

E.g.f.: exp(-t^2/4 + t/2)/sqrt(1-t). - Vladeta Jovovic, Aug 14 2006

a(n) ~ sqrt(2)*n^n/exp(n-1/4). - Vaclav Kotesovec, Oct 17 2012

Extensions

More terms from Alois P. Heinz, Sep 12 2008