OFFSET
0,2
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]
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
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
KEYWORD
nonn
AUTHOR
Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe
STATUS
approved