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A014507 Number of digraphs with loops, having unlabeled (non-isolated) nodes and n labeled edges.

%I A014507 #37 Jan 11 2024 19:25:39
%S A014507 1,2,13,162,3075,80978,2784067,119971162,6289972169,392257225754,
%T A014507 28582571639293,2398695602082442,229094801646110203,
%U A014507 24652935339990534970,2963620352166634246995,395067805289398293647026,58025593661340099613984593,9336949406574071339557552946
%N A014507 Number of digraphs with loops, having unlabeled (non-isolated) nodes and n labeled edges.
%D A014507 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A014507 Muniru A Asiru, <a href="/A014507/b014507.txt">Table of n, a(n) for n = 0..50</a>
%H A014507 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 A014507 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 A014507 a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(2*k). - _Vladeta Jovovic_, Jun 21 2003
%F A014507 E.g.f.: exp(-1)*Sum_{n>=0} (1+x)^(n^2)/n!. - _Paul D. Hanna_, Jul 03 2011
%F A014507 a(n) = n!*exp(-1)*Sum_{k>=sqrt(n)} binomial(k^2,n)/k!. - _Paul D. Hanna_, Jul 03 2011
%p A014507 A014507 := proc(n)
%p A014507     add(combinat[stirling1](n,k)*combinat[bell](2*k),k=0..n) ;
%p A014507 end proc:
%p A014507 seq(A014507(n),n=0..10) ; # _R. J. Mathar_, Apr 30 2017
%t A014507 a[n_] := Sum[StirlingS1[n, k]*BellB[2*k], {k, 0, n}];
%t A014507 Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, Jan 21 2018, from _Vladeta Jovovic_'s formula *)
%o A014507 (PARI) /* From _Vladeta Jovovic_'s formula: */
%o A014507 {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A014507 {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1),n)}
%o A014507 {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(2*k))}
%Y A014507 Cf. A000110 (Bell), A211250, A211251, A211252.
%K A014507 nonn
%O A014507 0,2
%A A014507 _Simon Plouffe_, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)