cp's OEIS Frontend

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

%I A020557 #54 Sep 08 2022 08:44:45
%S A020557 1,2,15,203,4140,115975,4213597,190899322,10480142147,682076806159,
%T A020557 51724158235372,4506715738447323,445958869294805289,
%U A020557 49631246523618756274,6160539404599934652455,846749014511809332450147,128064670049908713818925644
%N A020557 Number of oriented multigraphs on n labeled arcs (with loops).
%D A020557 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A020557 Vincenzo Librandi, <a href="/A020557/b020557.txt">Table of n, a(n) for n = 0..200</a>
%H A020557 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, Unpublished Doctoral thesis, City University London, 2013.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 A020557 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 A020557 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 A020557 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 A020557 a(n) = Bell(2*n) = A000110(2*n). - _Vladeta Jovovic_, Feb 02 2003
%F A020557 a(n) = exp(-1)*Sum_{k>=0} k^(2n)/k!. - _Benoit Cloitre_, May 19 2002
%F A020557 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 A020557 E.g.f.: exp(-1)*Sum_{n>=0}exp(n^2*x)/n!. - _Vladeta Jovovic_, Aug 24 2006
%t A020557 BellB[2 Range[0,20]] (* _Harvey P. Dale_, Jul 03 2021 *)
%o A020557 (PARI) for(n=0,50,print1(ceil(sum(i=0,1000,i^(2*n)/(i)!)/exp(1)),","))
%o A020557 (Sage) [bell_number(2*n) for n in range(0, 17)] # _Zerinvary Lajos_, May 14 2009
%o A020557 (Magma) [Bell(2*n): n in [0..20]]; // _Vincenzo Librandi_, Feb 05 2017
%o A020557 (Python)
%o A020557 from itertools import accumulate, islice
%o A020557 def A020557_gen(): # generator of terms
%o A020557     yield 1
%o A020557     blist, b = (1,), 1
%o A020557     while True:
%o A020557         for _ in range(2):
%o A020557             blist = list(accumulate(blist, initial=(b:=blist[-1])))
%o A020557         yield b
%o A020557 A020557_list = list(islice(A020557_gen(),30)) # _Chai Wah Wu_, Jun 22 2022
%Y A020557 Cf. A070906. Bisection of Bell numbers A000110.
%Y A020557 Cf. A099977.
%K A020557 nonn
%O A020557 0,2
%A A020557 Gilbert Labelle (gilbert(AT)lacim.uqam.ca), _Simon Plouffe_