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

Author

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

Keywords

• nonn

References

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

Links

• 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]

Programs

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 *)

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