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

Author

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

Keywords

• nonn

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• Elizabeth Banjo, Representation theory of algebras related to the partition algebra, Unpublished Doctoral thesis, City University London, 2013.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

Crossrefs

Cf. A070906. Bisection of Bell numbers A000110.

Cf. A099977.

Programs

Magma

[Bell(2*n): n in [0..20]]; // Vincenzo Librandi, Feb 05 2017

Mathematica

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

PARI

for(n=0,50,print1(ceil(sum(i=0,1000,i^(2*n)/(i)!)/exp(1)),","))

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

Sage

[bell_number(2*n) for n in range(0, 17)] # Zerinvary Lajos, May 14 2009

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