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A020561 Number of ordered oriented multigraphs on n labeled arcs (with loops). 1
1, 2, 17, 252, 5535, 165278, 6355147, 303080956, 17440307953, 1185613611362, 93640428880873, 8476453909912332, 869565923845396207, 100138764123162257470, 12840593975018953569971, 1820531766301308581051116, 283643668353734597645391393 (list; graph; refs; listen; history; text; internal format)
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

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

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: A099698 A218681 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

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Last modified September 21 20:10 EDT 2021. Contains 347598 sequences. (Running on oeis4.)