%I #54 Sep 08 2022 08:44:45

%S 1,2,15,203,4140,115975,4213597,190899322,10480142147,682076806159,

%T 51724158235372,4506715738447323,445958869294805289,

%U 49631246523618756274,6160539404599934652455,846749014511809332450147,128064670049908713818925644

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

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

%H Vincenzo Librandi, <a href="/A020557/b020557.txt">Table of n, a(n) for n = 0..200</a>

%H Elizabeth Banjo, <a href="http://openaccess.city.ac.uk/2360/1/Banjo,_Elizabeth.pdf">Representation theory of algebras related to the partition algebra</a>, Unpublished Doctoral thesis, City University London, 2013.

%H Laura Colmenarejo, Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki, <a href="https://arxiv.org/abs/1905.02071">An insertion algorithm on multiset partitions with applications to diagram algebras</a>, arXiv:1905.02071 [math.CO], 2019.

%H G. Labelle, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00265-4">Counting enriched multigraphs according to the number of their edges (or arcs)</a>, Discrete Math., 217 (2000), 237-248.

%H G. Paquin, <a href="/A038205/a038205.pdf">Dénombrement de multigraphes enrichis</a>, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

%F a(n) = Bell(2*n) = A000110(2*n). - _Vladeta Jovovic_, Feb 02 2003

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

%F 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.

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

%t BellB[2 Range[0,20]] (* _Harvey P. Dale_, Jul 03 2021 *)

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

%o (Sage) [bell_number(2*n) for n in range(0, 17)] # _Zerinvary Lajos_, May 14 2009

%o (Magma) [Bell(2*n): n in [0..20]]; // _Vincenzo Librandi_, Feb 05 2017

%o (Python)

%o from itertools import accumulate, islice

%o def A020557_gen(): # generator of terms

%o yield 1

%o blist, b = (1,), 1

%o while True:

%o for _ in range(2):

%o blist = list(accumulate(blist, initial=(b:=blist[-1])))

%o yield b

%o A020557_list = list(islice(A020557_gen(),30)) # _Chai Wah Wu_, Jun 22 2022

%Y Cf. A070906. Bisection of Bell numbers A000110.

%Y Cf. A099977.

%K nonn

%O 0,2

%A Gilbert Labelle (gilbert(AT)lacim.uqam.ca), _Simon Plouffe_