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A014507 Number of digraphs with loops, having unlabeled (non-isolated) nodes and n labeled edges. 19
1, 2, 13, 162, 3075, 80978, 2784067, 119971162, 6289972169, 392257225754, 28582571639293, 2398695602082442, 229094801646110203, 24652935339990534970, 2963620352166634246995 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 27 11:33 EDT 2021. Contains 348276 sequences. (Running on oeis4.)