A020565 - OEIS

%I #26 Jan 21 2018 04:08:39

%S 1,2,15,205,4202,118096,4300364,195155304,10727473182,698874420944,

%T 53040545101942,4624423933685370,457851029540848580,

%U 50977215595819988320,6329927203532081983976,870296461701522595081624,131659595370255359745290076

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

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

%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} (-1)^(n-k)*Stirling1(n, k)*A014507(k). - _Vladeta Jovovic_, May 02 2004

%F E.g.f.: Sum(Bell(2*n)*log(1-log(1-x))^n/n!, n=0..infinity). - _Vladeta Jovovic_, May 02 2004

%F E.g.f.: exp(-1)*Sum((1-log(1-x))^(n^2)/n!,n=0..infinity). - _Vladeta Jovovic_, Mar 04 2008

%p A020565 := proc(n)

%p     add((-1)^(n-k)*combinat[stirling1](n,k)*A014507(k),k=0..n) ;

%p end proc:

%p seq(A020565(n),n=0..10) ; # _R. J. Mathar_, Apr 30 2017

%t b[n_] := Sum[StirlingS1[n, k]*BellB[2*k], {k, 0, n}];

%t a[n_] := Sum[(-1)^(n-k)*StirlingS1[n, k]*b[k], {k, 0, n}];

%t Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jan 21 2018, after _Vladeta Jovovic_ *)

%K nonn

%O 0,2

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