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

0,4

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A000985, A002137.

Binomial transform of A001205.

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

Sequence in context: A234939 A365751 A192784 * A020031 A179323 A001340

Adjacent sequences: A108243 A108244 A108245 * A108247 A108248 A108249

Keyword

nonn

Author

Marni Mishna, Jun 17 2005

Extensions

More terms from Alois P. Heinz, Sep 12 2008

Status

approved