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A098623 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs.

%I A098623 #14 Jan 13 2021 01:20:53
%S A098623 1,1,8,109,2229,62684,2289151,104344153,5767234550,378073098155,
%T A098623 28888082263581,2536660090249102,253007765488793325,
%U A098623 28383529110762969901,3551558435250676339536,492092920443604792460905,75025155137863150912784409,12516480979952118669729618300
%N A098623 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs.
%D A098623 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A098623 Andrew Howroyd, <a href="/A098623/b098623.txt">Table of n, a(n) for n = 0..200</a>
%H A098623 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 A098623 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 A098623 E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014505 and 1 + R(x) is the e.g.f. of A000110. - _Andrew Howroyd_, Jan 12 2021
%o A098623 (PARI) \\ here R(n) is A000110 as e.g.f.
%o A098623 egfA020556(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i, k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
%o A098623 EnrichedGdSeq(R)={my(n=serprec(R, x)-1, B=subst(egfA020556(n), x, log(1+x + O(x*x^n)))); Vec(serlaplace(subst(B, x, R-polcoef(R,0))))}
%o A098623 R(n)={exp(exp(x + O(x*x^n))-1)}
%o A098623 EnrichedGdSeq(R(20)) \\ _Andrew Howroyd_, Jan 12 2021
%Y A098623 Cf. A000110, A098620, A098621, A098622.
%K A098623 nonn
%O A098623 0,3
%A A098623 _N. J. A. Sloane_, Oct 26 2004
%E A098623 Terms a(12) and beyond from _Andrew Howroyd_, Jan 12 2021