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.

1, 2, 17, 250, 5465, 162677, 6241059, 297132409, 17075153860, 1159545515804, 91501467848088, 8276847825732141, 848577193578286942, 97672164219292005480, 12518933902769241287267, 1774279753092963892540493, 276351502436571180980604240, 47046745370508674770872396843

Offset

0,2

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]

Formula

E.g.f.: exp(-1)*Sum_{n >=0} exp(n^2*(exp(x)-1))/n!. - Vladeta Jovovic, Aug 24 2006

a(n) = Sum_{k=0..n} Stirling2(n,k)*Bell(2*k). - Vladeta Jovovic, Aug 24 2006

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

Prog

(PARI) \\ here R(n) is A000110 as e.g.f.
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)}
EnrichedGdlSeq(R)={my(n=serprec(R, x)-1); Vec(serlaplace(subst(egfA014507(n), x, R-polcoef(R, 0))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

Crossrefs

Cf. A000110, A014507, A098620, A098621, A098623.

Sequence in context: A099698 A218681 A349293 * A020561 A099702 A253549

Adjacent sequences: A098619 A098620 A098621 * A098623 A098624 A098625

Keyword

nonn

Author

N. J. A. Sloane, Oct 26 2004

Extensions

More terms from Vladeta Jovovic, Aug 24 2006

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

Terms a(16) and beyond from Andrew Howroyd, Jan 12 2021

Status

approved