A014507 Number of digraphs with loops, having unlabeled (non-isolated) nodes and n labeled edges.

1, 2, 13, 162, 3075, 80978, 2784067, 119971162, 6289972169, 392257225754, 28582571639293, 2398695602082442, 229094801646110203, 24652935339990534970, 2963620352166634246995, 395067805289398293647026, 58025593661340099613984593, 9336949406574071339557552946

Offset

0,2

References

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

Links

Muniru A Asiru, Table of n, a(n) for n = 0..50

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

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(2*k). - Vladeta Jovovic, Jun 21 2003

E.g.f.: exp(-1)*Sum_{n>=0} (1+x)^(n^2)/n!. - Paul D. Hanna, Jul 03 2011

a(n) = n!*exp(-1)*Sum_{k>=sqrt(n)} binomial(k^2,n)/k!. - Paul D. Hanna, Jul 03 2011

Maple

A014507 := proc(n)
    add(combinat[stirling1](n, k)*combinat[bell](2*k), k=0..n) ;
end proc:
seq(A014507(n), n=0..10) ; # R. J. Mathar, Apr 30 2017

Mathematica

a[n_] := Sum[StirlingS1[n, k]*BellB[2*k], {k, 0, n}];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jan 21 2018, from Vladeta Jovovic's formula *)

Prog

(PARI) /* From Vladeta Jovovic's formula: */
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(2*k))}

Crossrefs

Cf. A000110 (Bell), A211250, A211251, A211252.

Sequence in context: A347051 A291140 A192563 * A132614 A187927 A252766

Adjacent sequences: A014504 A014505 A014506 * A014508 A014509 A014510

Keyword

nonn

Author

Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)

Status

approved