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A020565 Number of cyclic oriented multigraphs on n labeled arcs (with loops).

%I A020565 #26 Jan 21 2018 04:08:39
%S A020565 1,2,15,205,4202,118096,4300364,195155304,10727473182,698874420944,
%T A020565 53040545101942,4624423933685370,457851029540848580,
%U A020565 50977215595819988320,6329927203532081983976,870296461701522595081624,131659595370255359745290076
%N A020565 Number of cyclic oriented multigraphs on n labeled arcs (with loops).
%D A020565 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A020565 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 A020565 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 A020565 a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*A014507(k). - _Vladeta Jovovic_, May 02 2004
%F A020565 E.g.f.: Sum(Bell(2*n)*log(1-log(1-x))^n/n!, n=0..infinity). - _Vladeta Jovovic_, May 02 2004
%F A020565 E.g.f.: exp(-1)*Sum((1-log(1-x))^(n^2)/n!,n=0..infinity). - _Vladeta Jovovic_, Mar 04 2008
%p A020565 A020565 := proc(n)
%p A020565     add((-1)^(n-k)*combinat[stirling1](n,k)*A014507(k),k=0..n) ;
%p A020565 end proc:
%p A020565 seq(A020565(n),n=0..10) ; # _R. J. Mathar_, Apr 30 2017
%t A020565 b[n_] := Sum[StirlingS1[n, k]*BellB[2*k], {k, 0, n}];
%t A020565 a[n_] := Sum[(-1)^(n-k)*StirlingS1[n, k]*b[k], {k, 0, n}];
%t A020565 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jan 21 2018, after _Vladeta Jovovic_ *)
%K A020565 nonn
%O A020565 0,2
%A A020565 Gilbert Labelle (gilbert(AT)lacim.uqam.ca), _Simon Plouffe_