A099977 -id:A099977 - cp's OEIS frontend

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

Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe

nonn

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

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Elizabeth Banjo, Representation theory of algebras related to the partition algebra, Unpublished Doctoral thesis, City University London, 2013.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

Cf. A070906. Bisection of Bell numbers A000110.

Cf. A099977.

[Bell(2*n): n in [0..20]]; // Vincenzo Librandi, Feb 05 2017

BellB[2 Range[0,20]] (* Harvey P. Dale, Jul 03 2021 *)

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

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

[bell_number(2*n) for n in range(0, 17)] # Zerinvary Lajos, May 14 2009

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

A208301 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 5, 42, 52, 15, 15, 602, 2906, 877, 52, 52, 12840, 373780, 433252, 21147, 203, 203, 373780, 87852626, 656404264, 113503692, 678570, 877, 877, 14050312, 33093356640, 2227156082842, 2475181138384, 46538017584, 27644437, 4140

Offset: 1

R. H. Hardin, Feb 25 2012

			Table starts
....1..........1.................2.....................5
....2..........5................42...................602
....5.........52..............2906................373780
...15........877............433252.............656404264
...52......21147.........113503692.........2475181138384
..203.....678570.......46538017584.....17131843186425504
..877...27644437....27700815674032.196551307092757144384
.4140.1382958545.22702140562948192
Some solutions for n=4 k=3
..0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..2....0..1..0
..0..1..0....0..1..0....0..1..0....0..1..0....2..3..0....0..3..0....0..1..0
..0..1..0....0..2..1....0..1..0....0..1..0....0..2..0....0..1..0....2..1..0
..0..2..1....0..2..0....0..1..0....0..1..2....0..1..0....0..3..0....2..1..2

Andrew Howroyd, Table of n, a(n) for n = 1..210 (terms 1..58 from R. H. Hardin)

Columns 1..6 are A000110, A099977(n-1), A208297, A208298, A208299, A208300
Rows 1..7 are A000110(n-1), A208302, A208303, A208305, A208305, A208306, A208307.
Main diagonal is A361450.
Cf. A207868.

A308647 a(n) = exp(1) * Sum_{k>=0} (-1)^kk^(2n+1)/k!.

Original entry on oeis.org

-1, 1, -2, -9, 267, -2180, -50533, 1966797, 8638718, -2540956509, 27172288399, 5592543175252, -168392610536153, -20819319685262839, 1122009166836993406, 127595724180314195839, -9985347479130060737373, -1244077225312583088164916, 120225865637787689310572899, 18462990063073814590636032245

Offset: 0

Ilya Gutkovskiy, Jun 13 2019

sign

Eric Weisstein's World of Mathematics, Complementary Bell Number

Cf. A000587, A099977, A308646.

Table[Exp[1] Sum[(-1)^k k^(2 n + 1)/k!, {k, 0, Infinity}], {n, 0, 19}]
Table[BellB[2 n + 1, -1], {n, 0, 19}]

a(n) = Sum_{k=0..2*n+1} (-1)^k*Stirling2(2*n+1,k).

a(n) = A000587(2*n+1).

A308865 a(n) = Sum_{k>=0} k^(2*n+1)/2^(k+1).

Original entry on oeis.org

1, 13, 541, 47293, 7087261, 1622632573, 526858348381, 230283190977853, 130370767029135901, 92801587319328411133, 81124824998504073881821, 85438451336745709294580413, 106697365438475775825583498141, 155897763918621623249276226253693, 263478385263023690020893329044576861

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Cf. A000670, A008277, A099977, A249938.

Table[Sum[k^(2 n + 1)/2^(k + 1), {k, 0, Infinity}], {n, 0, 14}]
Table[Sum[k! StirlingS2[2 n + 1, k], {k, 0, 2 n + 1}], {n, 0, 14}]

a(n) = Sum_{k=0..2*n+1} k!*Stirling2(2*n+1,k).
a(n) = A000670(2*n+1).
a(n) ~ sqrt(Pi) * 2^(2*n + 1) * n^(2*n + 3/2) / (exp(2*n) * (log(2))^(2*n + 2)). - Vaclav Kotesovec, Sep 25 2019

cp's OEIS Frontend

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

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A208301 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or antidiagonal neighbor (colorings ignoring permutations of colors).

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A308647 a(n) = exp(1) * Sum_{k>=0} (-1)^kk^(2n+1)/k!.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A308865 a(n) = Sum_{k>=0} k^(2*n+1)/2^(k+1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Sage

Formula

A208301 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or antidiagonal neighbor (colorings ignoring permutations of colors).

Views

Author

Keywords

Examples

Links

Crossrefs

A308647 a(n) = exp(1) * Sum_{k>=0} (-1)^k*k^(2*n+1)/k!.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A308865 a(n) = Sum_{k>=0} k^(2*n+1)/2^(k+1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A308647 a(n) = exp(1) * Sum_{k>=0} (-1)^kk^(2n+1)/k!.