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A098622 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled loops and arcs.

%I A098622 #20 Jan 12 2021 21:29:58
%S A098622 1,2,17,250,5465,162677,6241059,297132409,17075153860,1159545515804,
%T A098622 91501467848088,8276847825732141,848577193578286942,
%U A098622 97672164219292005480,12518933902769241287267,1774279753092963892540493,276351502436571180980604240,47046745370508674770872396843
%N A098622 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled loops and arcs.
%D A098622 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A098622 Andrew Howroyd, <a href="/A098622/b098622.txt">Table of n, a(n) for n = 0..200</a>
%H A098622 G. Labelle, <a href="https://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 A098622 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 A098622 E.g.f.: exp(-1)*Sum_{n >=0} exp(n^2*(exp(x)-1))/n!. - _Vladeta Jovovic_, Aug 24 2006
%F A098622 a(n) = Sum_{k=0..n} Stirling2(n,k)*Bell(2*k). - _Vladeta Jovovic_, Aug 24 2006
%F A098622 E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000110. - _Andrew Howroyd_, Jan 12 2021
%o A098622 (PARI) \\ here R(n) is A000110 as e.g.f.
%o A098622 egfA014507(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, stirling(i,k,1)*polcoef(bell, 2*k))*x^i/i!) + O(x*x^n)}
%o A098622 EnrichedGdlSeq(R)={my(n=serprec(R, x)-1); Vec(serlaplace(subst(egfA014507(n), x, R-polcoef(R,0))))}
%o A098622 R(n)={exp(exp(x + O(x*x^n))-1)}
%o A098622 EnrichedGdlSeq(R(20)) \\ _Andrew Howroyd_, Jan 12 2021
%Y A098622 Cf. A000110, A014507, A098620, A098621, A098623.
%K A098622 nonn
%O A098622 0,2
%A A098622 _N. J. A. Sloane_, Oct 26 2004
%E A098622 More terms from _Vladeta Jovovic_, Aug 24 2006
%E A098622 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 15 2007
%E A098622 Terms a(16) and beyond from _Andrew Howroyd_, Jan 12 2021