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

0,2

References

G. Labelle, Counting enriched multigraphs..., Discrete Math., 217 (2000), 237-248.

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

Formula

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

Prog

(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 */

Crossrefs

Sequence in context: A218681 A349293 A098622 * A099702 A253549 A002590

Adjacent sequences: A020558 A020559 A020560 * A020562 A020563 A020564

Keyword

nonn

Author

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

Status

approved