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- Elizabeth Banjo, Representation theory of algebras related to the partition algebra, Unpublished Doctoral thesis, City University London, 2013.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