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A098620 Consider the family of multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled edges.

%I A098620 #13 Jan 12 2021 18:54:30
%S A098620 1,1,4,26,257,3586,66207,1540693,43659615,1469677309,57681784820,
%T A098620 2601121752854,133170904684965,7664254746784243,491679121677763607,
%U A098620 34905596059311761907,2725010800987216480527,232643959843709167832482,21613761720729431904201734
%N A098620 Consider the family of multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled edges.
%D A098620 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A098620 Andrew Howroyd, <a href="/A098620/b098620.txt">Table of n, a(n) for n = 0..200</a>
%H A098620 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 A098620 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 A098620 E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 and 1 + R(x) is the e.g.f. of A000110. - _Andrew Howroyd_, Jan 12 2021
%o A098620 (PARI) \\ here R(n) is A000110 as e.g.f.
%o A098620 egf1(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 A098620 EnrichedGnSeq(R)={my(n=serprec(R, x)-1, B=exp(x/2 + O(x*x^n))*subst(egf1(n), x, log(1+x + O(x*x^n))/2)); Vec(serlaplace(subst(B, x, R-polcoef(R,0))))}
%o A098620 R(n)={exp(exp(x + O(x*x^n))-1)}
%o A098620 EnrichedGnSeq(R(20)) \\ _Andrew Howroyd_, Jan 12 2021
%Y A098620 Cf. A000110, A014500, A098621, A098622, A098623.
%K A098620 nonn
%O A098620 0,3
%A A098620 _N. J. A. Sloane_, Oct 26 2004
%E A098620 Terms a(12) and beyond from _Andrew Howroyd_, Jan 12 2021