A020561 Number of ordered oriented multigraphs on n labeled arcs (with loops).

1, 2, 17, 252, 5535, 165278, 6355147, 303080956, 17440307953, 1185613611362, 93640428880873, 8476453909912332, 869565923845396207, 100138764123162257470, 12840593975018953569971, 1820531766301308581051116, 283643668353734597645391393

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

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200
• 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

PARI

/* From Vladeta Jovovic's formula: */
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, abs(Stirling1(n, k))*Bell(2*k))}

PARI

{a(n)=round(n!*exp(-1)*suminf(k=0, binomial(k^2 + n-1,n)/k!))} /* Paul D. Hanna */

Formula

a(n) = Sum_{k=0..n} abs(Stirling1(n, k))*Bell(2*k). - Vladeta Jovovic, Jun 21 2003

E.g.f.: exp(-1)*Sum_{n>=0} (1-x)^(-n^2)/n!. - Paul D. Hanna, Jul 03 2011

a(n) = n!*exp(-1)*Sum_{k>=0} binomial(k^2 + n-1,n)/k!. - Paul D. Hanna, Jul 03 2011