%I #31 Aug 09 2024 16:21:08

%S 1,2,17,252,5535,165278,6355147,303080956,17440307953,1185613611362,

%T 93640428880873,8476453909912332,869565923845396207,

%U 100138764123162257470,12840593975018953569971,1820531766301308581051116,283643668353734597645391393

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

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

%H Vaclav Kotesovec, <a href="/A020561/b020561.txt">Table of n, a(n) for n = 0..200</a>

%H G. Labelle, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00265-4">Counting enriched multigraphs according to the number of their edges (or arcs)</a>, Discrete Math., 217 (2000), 237-248.

%H G. Paquin, <a href="/A038205/a038205.pdf">Dénombrement de multigraphes enrichis</a>, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

%F a(n) = Sum_{k=0..n} abs(Stirling1(n, k))*Bell(2*k). - _Vladeta Jovovic_, Jun 21 2003

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

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

%o (PARI) /* From _Vladeta Jovovic_'s formula: */

%o {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

%o {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}

%o {a(n)=sum(k=0, n, abs(Stirling1(n, k))*Bell(2*k))}

%o (PARI) {a(n)=round(n!*exp(-1)*suminf(k=0, binomial(k^2 + n-1,n)/k!))} /* _Paul D. Hanna_ */

%K nonn

%O 0,2

%A Gilbert Labelle (gilbert(AT)lacim.uqam.ca), _Simon Plouffe_