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Elizabeth Banjo, Representation theory of algebras related to the partition algebra, Unpublished Doctoral thesis, City University London, 2013.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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe
STATUS
approved