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A020557
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Number of oriented multigraphs on n labeled arcs (with loops).
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18
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1, 2, 15, 203, 4140, 115975, 4213597, 190899322, 10480142147, 682076806159, 51724158235372, 4506715738447323, 445958869294805289, 49631246523618756274, 6160539404599934652455, 846749014511809332450147, 128064670049908713818925644
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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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Elizabeth Banjo, Representation theory of algebras related to the partition algebra, Unpublished Doctoral thesis, City University London, 2013.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]
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FORMULA
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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
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MATHEMATICA
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BellB[2 Range[0, 20]] (* Harvey P. Dale, Jul 03 2021 *)
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PROG
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(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
A020557_list = list(islice(A020557_gen(), 30)) # Chai Wah Wu, Jun 22 2022
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CROSSREFS
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Cf. A070906. Bisection of Bell numbers A000110.
Cf. A099977.
Sequence in context: A221102 A351920 A319466 * A323118 A184361 A351501
Adjacent sequences: A020554 A020555 A020556 * A020558 A020559 A020560
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KEYWORD
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nonn
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AUTHOR
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Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe
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STATUS
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approved
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