A020557 Number of oriented multigraphs on n labeled arcs (with loops).

1, 2, 15, 203, 4140, 115975, 4213597, 190899322, 10480142147, 682076806159, 51724158235372, 4506715738447323, 445958869294805289, 49631246523618756274, 6160539404599934652455, 846749014511809332450147, 128064670049908713818925644

Offset

0,2

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Elizabeth Banjo, Representation theory of algebras related to the partition algebra, Unpublished Doctoral thesis, City University London, 2013.

Laura Colmenarejo, Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki, An insertion algorithm on multiset partitions with applications to diagram algebras, arXiv:1905.02071 [math.CO], 2019.

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) = Bell(2*n) = A000110(2*n). - Vladeta Jovovic, Feb 02 2003

a(n) = exp(-1)*Sum_{k>=0} k^(2n)/k!. - Benoit Cloitre, May 19 2002

E.g.f.: exp(x*(d_z)^2)*(exp(exp(z)-1))|_{z=0}, with the derivative operator d_z := d/dz. Adapted from eqs.(14) and (15) of the 1999 C. M. Bender reference given in A000110.

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

Mathematica

BellB[2 Range[0, 20]] (* Harvey P. Dale, Jul 03 2021 *)

Prog

(PARI) for(n=0, 50, print1(ceil(sum(i=0, 1000, i^(2*n)/(i)!)/exp(1)), ", "))
(Sage) [bell_number(2*n) for n in range(0, 17)] # Zerinvary Lajos, May 14 2009
(Magma) [Bell(2*n): n in [0..20]]; // Vincenzo Librandi, Feb 05 2017
(Python)
from itertools import accumulate, islice
def A020557_gen(): # generator of terms
    yield 1
    blist, b = (1, ), 1
    while True:
        for _ in range(2):
            blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b
A020557_list = list(islice(A020557_gen(), 30)) # Chai Wah Wu, Jun 22 2022

Crossrefs

Cf. A070906. Bisection of Bell numbers A000110.

Cf. A099977.

Sequence in context: A221102 A351920 A319466 * A323118 A184361 A351501

Adjacent sequences: A020554 A020555 A020556 * A020558 A020559 A020560

Keyword

nonn

Author

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

Status

approved