A020565 Number of cyclic oriented multigraphs on n labeled arcs (with loops).

1, 2, 15, 205, 4202, 118096, 4300364, 195155304, 10727473182, 698874420944, 53040545101942, 4624423933685370, 457851029540848580, 50977215595819988320, 6329927203532081983976, 870296461701522595081624, 131659595370255359745290076

Offset

0,2

References

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Links

Table of n, a(n) for n=0..16.

G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*A014507(k). - Vladeta Jovovic, May 02 2004

E.g.f.: Sum(Bell(2*n)*log(1-log(1-x))^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004

E.g.f.: exp(-1)*Sum((1-log(1-x))^(n^2)/n!,n=0..infinity). - Vladeta Jovovic, Mar 04 2008

Maple

A020565 := proc(n)
    add((-1)^(n-k)*combinat[stirling1](n, k)*A014507(k), k=0..n) ;
end proc:
seq(A020565(n), n=0..10) ; # R. J. Mathar, Apr 30 2017

Mathematica

b[n_] := Sum[StirlingS1[n, k]*BellB[2*k], {k, 0, n}];
a[n_] := Sum[(-1)^(n-k)*StirlingS1[n, k]*b[k], {k, 0, n}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jan 21 2018, after Vladeta Jovovic *)

Crossrefs

Sequence in context: A184361 A351501 A124558 * A282521 A099718 A143881

Adjacent sequences: A020562 A020563 A020564 * A020566 A020567 A020568

Keyword

nonn

Author

Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe

Status

approved