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

1, 1, 4, 26, 257, 3586, 66207, 1540693, 43659615, 1469677309, 57681784820, 2601121752854, 133170904684965, 7664254746784243, 491679121677763607, 34905596059311761907, 2725010800987216480527, 232643959843709167832482, 21613761720729431904201734

Offset: 0

Author

N. J. A. Sloane, Oct 26 2004

Keywords

• nonn

References

• G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
• G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Crossrefs

Cf. A000110, A014500, A098621, A098622, A098623.

Programs

PARI

\\ here R(n) is A000110 as e.g.f.
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)}
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))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Formula

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

Extensions

Terms a(12) and beyond from Andrew Howroyd, Jan 12 2021