%I A000110 M1484 N0585
%S A000110 1,1,2,5,15,52,203,877,4140,21147,115975,678570,4213597,27644437,
%T A000110 190899322,1382958545,10480142147,82864869804,682076806159,
%U A000110 5832742205057,51724158235372,474869816156751,4506715738447323
%N A000110 Bell or exponential numbers: ways of placing n labeled balls into n indistinguishable
boxes.
%C A000110 Number of partitions of a set of n labeled elements.
%C A000110 a(n-1) = number of nonisomorphic colorings of a map consisting of a row
of n+1 adjacent regions. - David W. Wilson, Feb 22, 2005
%C A000110 If an integer is square-free and has n distinct prime factors then a(n)
is the number of ways of writing it as a product of its divisors
- Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 23 2001
%C A000110 Consider rooted trees of height at most 2. Letting each tree 'grow' into
the next generation of n means we produce a new tree for every node
which is either the root or at height 1, which gives the Bell numbers.
- Jon Perry (perry(AT)globalnet.co.uk), Jul 23 2003
%C A000110 Begin with [1,1] and follow the rule that [1,k] -> [1,k+1] and [1,k]
k times, e.g. [1,3] is transformed to [1,4], [1,3], [1,3], [1,3].
Then a(n) is the sum of all components. [1,1]=2, [1,2],[1,1]=5, [1,
3],[1,2],[1,2],[1,1],[1,2]=15, etc... - Jon Perry (perry(AT)globalnet.co.uk),
Mar 05 2004
%C A000110 Number of distinct rhyme schemes for a poem of n lines: a rhyme scheme
is a string of letters (eg, 'abba') such that the leftmost letter
is always 'a' and no letter may be greater than one more than the
greatest letter to its left. Thus 'aac' is not valid since 'c' is
more than one greater than 'a'. For example, a(3)=5 because there
are 5 rhyme schemes. aaa, aab, aba, abb, abc. - Bill Blewett (BillBle(AT)microsoft.com),
Mar 23 2004
%C A000110 Comment from Neven Juric, Oct 19 2009: (Start)
%C A000110 The case n=4: aaaa, aaab, aaba, aabb, aabc, abaa, abab, abac, abba, abbb,
abbc, abca, abcb, abcc, abcd
%C A000110 The case n=5:
%C A000110 aaaaa, aaaab, aaaba, aaabb, aaabc, aabaa, aabab, aabac, aabba, aabbb,
%C A000110 aabbc, aabca, aabcb, aabcc, aabcd, abaaa, abaab, abaac, ababa, ababb,
%C A000110 ababc, abaca, abacb, abacc, abacd, abbaa, abbab, abbac, abbba, abbbb,
%C A000110 abbbc, abbca, abbcb, abbcc, abbcd, abcaa, abcab, abcac, abcad, abcba,
%C A000110 abcbb, abcbc, abcbd, abcca, abccb, abccc, abccd, abcda, abcdb, abcdc,
%C A000110 abcdd, abcde (End)
%C A000110 Also the number of equivalence relations in (alternatively, or the number
of partitions of) a set of n elements. - Federico Arboleda (federico.arboleda(AT)gmail.com),
Mar 09 2005
%C A000110 Number of partitions of {1, ...,n+1} into subsets of nonconsecutive integers,
including the partition 1|2|...|n+1. E.g. a(3)=5: there are 5 partitions
of {1,2,3,4} into subsets of nonconsecutive integers namely 13|24,
13|2|4, 14|2|3, 1|24|3, 1|2|3|4. - A. O. Munagi (amunagi(AT)yahoo.com),
Mar 20 2005
%C A000110 Triangle (addition) scheme to produce terms, derived from the recurrence,
from Oscar Arevalo (loarevalo(AT)sbcglobal.net), May 11 2005:
%C A000110 1
%C A000110 1 2
%C A000110 2 3 5
%C A000110 5 7 10 15
%C A000110 15 20 27 37 ... [This is Aitken's array A011971]
%C A000110 With p(n) = the number of integer partitions of n, p(i) = the number
of parts of the i-th partition of n, d(i) = the number of different
parts of the i-th partition of n, p(j,i) = the j-th part of the i-th
partition of n, m(i,j) = multiplicity of the j-th part of the i-th
partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)}
= product over j one has: a(n)=sum_{i=1}^{p(n)} (n!/(prod_{j=1}^{p(i)}p(i,
j)!)) * (1/(prod_{j=1}^{d(i)} m(i,j)!)) - Thomas Wieder (wieder.thomas(AT)t-online.de),
May 18 2005
%C A000110 a(n+1) = the number of binary relations on an n-element set that are
both symmetric and transitive. - Justin Witt (justinmwitt(AT)gmail.com),
Jul 12 2005
%C A000110 If Jon Perry's rule is used, i.e. "Begin with [1,1] and follow the rule
that [1,k] -> [1,k+1] and [1,k] k times, e.g. [1,3] is transformed
to [1,4], [1,3], [1,3], [1,3]. Then a(n) is the sum of all components.
[1,1]=2, [1,2],[1,1]=5, [1,3],[1,2],[1,2],[1,1],[1,2]=15, etc..."
then a(n-1) = [number of components used to form a(n)] / 2 - Daniel
Kuan (dkcm(AT)yahoo.com), Feb 19 2006
%C A000110 a(n) is the number of functions f from {1,...,n} to {1,...,n,n+1} that
satisfy the following two conditions for all x in the domain: (1)
f(x)>x; (2)f(x)=n+1 or f(f(x))=n+1.E.g. a(3)=5 because there are
exactly five functions that satisfy the two conditions: f1={(1,4),
(2,4),(3,4)}, f2={(1,4),(2,3),(3,4)}, f3={(1,3),(2,4),(3,4)}, f4={(1,
2),(2,4),(3,4)} and f5={(1,3),(2,3),(3,4)}. - Dennis P. Walsh (dwalsh(AT)mtsu.edu),
Feb 20 2006
%C A000110 Number of asynchronic siteswap patterns of length n which have no zero-throws
(i.e. contain no 0's) and whose number of orbits (in the sense given
by Allen Knutson) is equal to the number of balls. E.g. for n=4 the
condition is satisfied by the following 15 siteswaps 4444, 4413,
4242, 4134, 4112, 3441, 2424, 1344, 2411, 1313, 1241, 2222, 3131,
1124, 1111. Also number of ways to choose n permutations from identity
and cyclic permutations (1 2), (1 2 3), ..., (1 2 3 ... n) so that
their composition is identity. For n=3 we get the following five:
id o id o id, id o (1 2) o (1 2), (1 2) o id o (1 2), (1 2) o (1
2) o id, (1 2 3) o (1 2 3) o (1 2 3). (To see the bijection, look
at Ehrenborg and Readdy paper.) - Antti Karttunen (his-firstname.his-surname(AT)gmail.com),
May 01 2006.
%C A000110 a(n) = number of permutations on [n] in which a 3-2-1 (scattered) pattern
occurs only as part of a 3-2-4-1 pattern. Example. a(3) = 5 counts
all permutations on [3] except 321. See "Eigensequence for Composition"
ref. a(n) = number of permutation tableaux of size n (A000142) whose
first row contains no 0's. Example: a(3)=5 counts {{}, {}, {}}, {{1},
{}}, {{1}, {0}}, {{1}, {1}}, {{1, 1}}. - David Callan (callan(AT)stat.wisc.edu),
Oct 07 2006
%C A000110 Take the series 1^n/1! + 2^n/2! + 3^n/3! + 4^n/4! ... If n=1 then the
result will be e, about 2.71828. If n=2, the result will be 2e. If
n=3, the result will be 5e. This continues, following the pattern
of the Bell numbers: e, 2e, 5e, 15e, 52e, 203e, etc. - Jonathan R.
Love (japanada11(AT)yahoo.ca), Feb 22 2007
%C A000110 Comment from Gottfried Helms (helms(AT)uni-kassel.de), Mar 30 2007. (Start)
This sequence is also the first column in the matrix-exponential
of the (lower triangular) Pascal-matrix, scaled by exp(-1): PE =
exp(P) / exp(1) =
%C A000110 ....1...............................
%C A000110 ....1......1........................
%C A000110 ....2......2......1.................
%C A000110 ....5......6......3.....1...........
%C A000110 ...15.....20.....12.....4.....1.....
%C A000110 ...52.....75.....50....20.....5....1
%C A000110 ..203....312....225...100....30....6
%C A000110 ..877...1421...1092...525...175...42
%C A000110 First 4 columns are A000110, A033306, A105479, A105480. The general case
is mentioned in the two latter entries. PE is also the Hadamard-product
Toeplitz(A000110) (X) P:
%C A000110 ....1.........................
%C A000110 ....1.....1..................
%C A000110 ....2.....1....1.............
%C A000110 ....5.....2....1....1........
%C A000110 ...15.....5....2....1...1.... (X) P
%C A000110 ...52....15....5....2...1...1
%C A000110 ..203....52...15....5...2...1
%C A000110 ..877...203...52...15...5...2 (End)
%C A000110 The terms can also be computed with finite steps and precise integer
arithmetic. Instead of exp(P)/exp(1) one can compute A = exp(P -
I) where I is the identity-matrix of appropriate dimension since
(P-I) is nilpotent to the order of its dimension. Then a(n)=A[n,1]
where n is the row-index starting at 1. - Gottfried Helms helms(at)uni-kassel.de,
Apr 10 2007.
%C A000110 Comment from David W. Wilson (davidwwilson(AT)comcast.net), Aug 04 2007
and Sep 24 2007: Define a Bell pseudoprime to be a composite number
n such that a(n) == 2 (mod n). W. F. Lunnon recently found the Bell
pseudoprimes 21361 = 41*521 and C46 = 3*23*16218646893090134590535390526854205539989357
and conjectured that Bell pseudoprimes are extremely scarce. So the
second Bell pseudoprime is unlikely to be known with certainty in
the near future. I confirmed that 21361 is the first.
%C A000110 This sequence and A000587 form a reciprocal pair under the list partition
transform described in A133314. - Tom Copeland (tcjpn(AT)msn.com),
Oct 21 2007
%C A000110 Starting (1, 2, 5, 15, 52,...), equals row sums and right border of triangle
A136789. Also row sums of triangle A136790. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Jan 21 2008
%C A000110 This is the exp transform of A000012. [From Thomas Wieder (thomas.wieder(AT)t-online.de),
Sep 09 2008]
%C A000110 Contribution from A. Umar (aumarh(AT)squ.edu.om), Oct 12 2008: (Start)
%C A000110 a(n) is also the number of idempotent order-decreasing full transformations
(of an n-chain).
%C A000110 a(n) is also the number of nilpotent partial one-one order-decreasing
transformations (of an n-chain).
%C A000110 a(n+1) is also the number of partial one-one order-decreasing transformations
(of an n-chain). (End)
%C A000110 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 19 2008:
(Start)
%C A000110 Bell(n) is the number of n-pattern sequences [Cooper & Kennedy]. An n-pattern
sequence is a sequence of integers (a_1,...,a_n) such that a_i =
i or a_i = a_j for some j < i. For example, Bell(3) = 5 since the
3-pattern sequences are (1,1,1), (1,1,3), (1,2,1), (1,2,2) and (1,
2,3).
%C A000110 Bell(n) is the number of sequences of positive integers (N_1,...,N_n)
of length n such that N_1 = 1 and N_(i+1) <= 1 + max{j = 1..i} N_j
for i >= 1 (see the comment by B. Blewett above). It is interesting
to note that if we strengthen the latter condition to N_(i+1) <=
1 + N_i we get the Catalan numbers A000108 instead of the Bell numbers.
%C A000110 (End)
%C A000110 Equals the eigensequence of Pascal's triangle, A007318; and starting
with offset 1, = row sums of triangles A074664 and A152431. [From
Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 04 2008]
%C A000110 Contribution from David Pasino (davpas(AT)charter.net), Dec 04 2008:
(Start)
%C A000110 The entries f(i, j) in the exponential of the infinite lower-triangular
matrix
%C A000110 of binomial coefficients b(i, j) are f(i, j) = b(i, j) e a(i - j). (End)
%C A000110 Equals Lim_{k->inf.} A071919^k [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Jan 02 2009]
%C A000110 Equals A154107 convolved with A014182, where A014182 = expansion of exp(1-x-exp(-x)),
the eigensequence of A007318^(-1). Starting with offset 1 = A154108
convolved with (1,2,3,...) = row sums of triangle A154109. [From
Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 04 2009]
%C A000110 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 14 2009:
(Start)
%C A000110 Repeated iterates of (binomial transform of [1,0,0,0,...]) will converge
upon
%C A000110 (1, 2, 5, 15, 52,...) when each result is prefaced with a "1"; such that
the
%C A000110 final result is the fixed limit: (binomial transform of [1,1,2,5,15,...]
= (1,2,5,15,52,...). (End)
%C A000110 Contribution from Karol A. Penson (penson(AT)lptl.jussieu.fr), May 03
2009: (Start)
%C A000110 Relation between the Bell numbers B(n) and the nth derivative of 1/Gamma(1+x)
%C A000110 of such derivatives through seq(subs(x=0, simplify(diff(GAMMA(1+x)^(-1),
x$n))), n=1..6);
%C A000110 b) leave them expressed in terms of digamma (Psi(k))
%C A000110 and polygamma (Psi(k,n)) functions and unevaluated ;
%C A000110 Examples of such expressions, for n=1..5, are : n=1: -Psi(1), n=2: -(-Psi(1)^2+Psi(1,
1)),
%C A000110 n=3: -Psi(1)^3+3*Psi(1)*Psi(1,1)-Psi(2,1),
%C A000110 n=4: -(-Psi(1)^4+6*Psi(1)^2*Psi(1,1)-3*Psi(1,1)^2-4*Psi(1)*Psi(2,1)+Psi(3,
1)),
%C A000110 n=5: -Psi(1)^5+10*Psi(1)^3*Psi(1,1)-15*Psi(1)*Psi(1,1)^2-10*Psi(1)^2*Psi(2,
1)
%C A000110 +10*Psi(1,1)*Psi(2,1)+5*Psi(1)*Psi(3,1)-Psi(4,1) ;
%C A000110 c) for a given n read off the sum of absolute values
%C A000110 of coefficients of every term involving digamma or polygamma functions.
%C A000110 This sum is equal to B(n). Examples : B(1)=1, B(2)=1+1=2, B(3)=1+3+1=5,
B(4)=1+6+3+4+1=15, B(5)=1+10+15+10+10+5+1=52;
%C A000110 d) Observe that this decomposition of the Bell number B(n) apparently
does not involve the Stirling numbers of the second kind explicitly.
(End)
%C A000110 The numbers given above by Penson lead to the multinomial coefficients
A036040. - Johannes W. Meijer, Aug 14 2009
%C A000110 Column 1 of A162663. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net),
Jul 09 2009]
%C A000110 Asymptotic expansions (0!+1!+2!+...+(n-1)!)/(n-1)! = a(0) + a(1)/n +
a(2)/n^2 + ... and (0!+1!+2!+...+n!)/n! = 1 + a(0)/n + a(1)/n^2 +
a(2)/n^3 + ... - Michael Somos, Jun 28 2009
%C A000110 Starting with offset 1 = row sums of triangle A165194 [From Gary W. Adamson
(qntmpkt(AT)yahoo.com), Sep 06 2009]
%C A000110 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2009:
(Start)
%C A000110 a(n+1) = A165196(2^n); where A165196 begins: (1, 2, 4, 5, 7, 12, 14,
15,...).
%C A000110 such that A165196(2^3) = 15 = A000110(4). (End)
%C A000110 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16
2009: (Start)
%C A000110 The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ...
, m=>-1, which for m=-1 dates back to Euler, is related to the Bell
numbers. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1)
* A073003). We observe that A073003 is Gompertz's constant and that
A040027 was published by Gould, see for more information A163940.
%C A000110 (End)
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%H A000110 S. Plouffe, <a href="b000110.txt">Table of n, a(n) for n = 0..500</a>
%H A000110 K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon,
<a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives:
Dobinski relations and combinatorial identities</a>, J. Math. Phys.
vol. 50, 083512 (2009)
%H A000110 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000110 R. Aldrovandi and L. P. Freitas, <a href="http://arXiv.org/abs/physics/
9712026">Continuous iteration of dynamical maps</a>
%H A000110 Pat Ballew, <a href="http://www.pballew.net/Bellno.html">Bell Numbers</
a>
%H A000110 D. Barsky, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05barsky.html">
Analyse p-adique et suites classiques de nombres</a>, Sem. Loth.
Comb. B05b (1981) 1-21.
%H A000110 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/
abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized
Bell Numbers</a>
%H A000110 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/
abs/quant-ph/0402027">The general boson normal ordering problem.</
a>
%H A000110 H. Bottomley, <a href="A000110.gif">Illustration of initial terms</a>
%H A000110 A. Burstein and I. Lankham, <a href="http://arXiv.org/abs/math.CO/0506358">
Combinatorics of patience sorting piles</a>
%H A000110 David Callan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Callan/
callan96.html">A Combinatorial Interpretation of the Eigensequence
for Composition</a>.
%H A000110 David Callan, <a href="http://www.stat.wisc.edu/~callan/">Cesaro's Integral
Formula for the Bell Numbers (Corrected)</a>.
%H A000110 David Callan, <a href="http://arXiv.org/abs/0708.3301">Cesaro's integral
formula for the Bell numbers (corrected)</a>.
%H A000110 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000110 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer
Sequences, 4 (2001), #01.1.6.
%H A000110 A. Claesson and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0110036">
Counting patterns of type (1,2) or (2,1)</a>.
%H A000110 C. Cooper and R. E. Kennedy, <a href="http://www.math-cs.ucmo.edu/~curtisc/
articles.html">Patterns, automata and Stirling numbers of the second
kind</a>, Mathematics and Computer Education Journal, 26 (1992),
120-124. [From Peter Bala (pbala(AT)toucansurf.com), Oct 19 2008]
%H A000110 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/bell.html">
Bell number diagrams</a>
%H A000110 R. Ehrenborg and M. Readdy, <a href="http://www.ms.uky.edu/~readdy/Papers/
juggling.ps.gz">Juggling and applications to q-analogues</a>, Discrete
Math. 157 (1996), 107-125.
%H A000110 John Fiorillo, <a href="http://spectacle.berkeley.edu/~fiorillo/7genjimon.html">
GENJI-MON</a>
%H A000110 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
Publications/books.html">Analytic Combinatorics</a>, 2009; see page
109, 110
%H A000110 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/
k1bell.html">The Bell Numbers</a>
%H A000110 Daniel L. Geisler, <a href="http://www.tetration.org/Combinatorics/index.html">
Combinatorics of Iterated Functions</a>
%H A000110 A. Gertsch and A. M.Robert, <a href="http://www.kurims.kyoto-u.ac.jp/
EMIS/journals/BBMS/Bulletin/bul964/Robert-Gertsch.pdf">Some congruences
concerning the Bell numbers</a>
%H A000110 Gottfried Helms, <a href="http://go.helms-net.de/math/binomial/04_5_SummingBellStirling.pdf">
Bell Numbers</a>, 2008.
%H A000110 A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon,
<a href="http://arXiv.org/abs/quant-ph/0409152">A product formula
and combinatorial field theory</a>
%H A000110 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=15">
Encyclopedia of Combinatorial Structures 15</a>
%H A000110 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=65">
Encyclopedia of Combinatorial Structures 65</a>
%H A000110 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=73">
Encyclopedia of Combinatorial Structures 73</a>
%H A000110 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=291">
Encyclopedia of Combinatorial Structures 291</a>
%H A000110 A. Knutson, <a href="http://www.juggling.org/bin/mfs/JIS/help/siteswap/
faq.html#back">Siteswap FAQ, Section 5, Working backwards</a>, defines
the term "orbit" in siteswap notation.
%H A000110 Kazuhiro Kunii, <a href="http://plaza27.mbn.or.jp/~921/kumiko/genjiko/
genjikou.html">Genji-koh no zu</a> [Japanese page illustrating a(5)
= 52]
%H A000110 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
Sobalian Coefficients</a>.
%H A000110 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
index.html">Miscellaneous</a>.
%H A000110 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
On generalizations of Stirling number triangles</a>, J. Integer Seqs.,
Vol. 3 (2000), #00.2.4.
%H A000110 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
The Hankel Transform and Some of its Properties</a>, J. Integer Sequences,
4 (2001), #01.1.5.
%H A000110 R. J. Marsh and P. P. Martin, <a href="http://arXiv.org/abs/math.CO/0612572">
Pascal arrays: counting Catalan sets</a>
%H A000110 A. O. Munagi, <a href="http://www.hindawi.com/getarticle.aspx?doi=10.1155/
IJMMS.2005.215">k-Complementing Subsets of Nonnegative Integers</
a>, International Journal of Mathematics and Mathematical Sciences,
2005:2 (2005), 215-224.
%H A000110 A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L.
Graham et al., eds., Handbook of Combinatorics, 1995; see Examples
5.4 and 12.2. (<a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf">
pdf</a>, <a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.ps">
ps</a>)
%H A000110 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a
href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type
relations via substitution and the moment problem</a>
%H A000110 K. A. Penson and J.-M. Sixdeniers, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Integral Representations of Catalan and
Related Numbers</a>, J. Integer Sequences, 4 (2001), #01.2.5.
%H A000110 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting
Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004),
Article 04.3.2.
%H A000110 S. Plouffe, <a href="a000110.txt">Table of n, a(n) for n = 0..3015</a>
%H A000110 S. Plouffe, <a href="http://pi.lacim.uqam.ca/piDATA/bell.txt">Bell numbers
(first 1000 terms)</a>
%H A000110 T. Prellberg, <a href="http://algo.inria.fr/seminars/sem02-03/prellberg1-slides.ps">
On the asymptotic analysis of a class of linear recurrences</a> (slides).
%H A000110 C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">
Determinants de Hankel et theoreme de Sylvester</a>
%H A000110 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/NoteBooks/
NoteBook2/chapterIII/page5.htm">Notebook entry</a>
%H A000110 A. Ross, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
BellNumber.html">Bell number</a>
%H A000110 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/
series005">Bell numbers</a>
%H A000110 A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, <a
href="http://arXiv.org/abs/quant-ph/0310174">Combinatorial physics,
normal order and model Feynman graphs</a>.
%H A000110 A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, <a
href="http://arXiv.org/abs/quant-ph/0409082">Partition functions
and graphs: A combinatorial approach</a>.
%H A000110 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BellNumber.html">Link to a section of The World of Mathematics (1).</
a>
%H A000110 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BinomialTransform.html">Link to a section of The World of Mathematics
(2).</a>
%H A000110 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
StirlingTransform.html">Link to a section of The World of Mathematics
(3).</a>
%H A000110 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BellTriangle.html">Bell Triangle</a>
%H A000110 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</
a>, 2nd edn., Academic Press, NY, 1994, pp. 21ff.
%H A000110 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to
rooted trees</a>
%H A000110 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000110 <a href="Sindx_J.html#Juggling">Index entries for sequences related to
juggling</a>
%H A000110 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%F A000110 E.g.f.: exp (exp(x)- 1). Recurrence: a(n+1) = Sum a(k)C(n, k). Also a(n)
= Sum Stirling2(n, k), k=1..n.
%F A000110 a(n) = SUM(j = 0 to n-1, (1/(n-1)!) * A000166(j) * C(n-1, j) * (n-j)^(n-1)).
- Andre F. Labossiere (boronali(AT)laposte.net), Dec 01 2004
%F A000110 G.f.: sum(1/((1-k*x)*k!), k = 0 .. infinity)/exp(1) = hypergeom([ -1/
x], [(x-1)/x], 1)/exp(1)=((1-2*x)+LaguerreL(1/x, (x-1)/x, 1)+x*LaguerreL(1/
x, (2*x-1)/x, 1))*Pi/(x^2*sin(Pi*(2*x-1)/x)), where LaguerreL(mu,
nu, z) =( GAMMA(mu+nu+1)/GAMMA(mu+1)/GAMMA(nu+1))* hypergeom([ -mu],
[nu+1], z) is the Laguerre function, the analytic extension of the
Laguerre polynomials, for mu not equal to a nonnegative integer.
This generating function has an infinite number of poles accumulating
in the neighborhood of x=0.- Karol A. Penson (penson(AT)lptl.jussieu.fr),
Mar 25, 2002.
%F A000110 a(n) = exp(-1)*sum(k=>0, k^n/k!) [Dobinski] - Benoit Cloitre (benoit7848c(AT)orange.fr),
May 19 2002
%F A000110 a(n) is asymptotic to n!*(2 Pi r^2 exp(r))^(-1/2) exp(exp(r)-1) / r^n,
where r is the positive root of r exp(r) = n. - see e.g. the Odlyzko
reference.
%F A000110 a(n) is asymptotic to b^n*exp(b-n-1/2)/sqrt(ln(n)) where b satisfies
b*ln(b) = n - 1/2 (see Graham, Knuth and Patashnik, Concrete Mathematics,
2nd ed., p. 493) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct
23 2002
%F A000110 G.f.: sum{k>=0, x^k/prod[l=1..k, 1-lx]}. - R. Stephan, Apr 18 2004
%F A000110 a(n+1) = exp(-1)*sum(k=>0, (k+1)^(n)/k!) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net),
Jun 03 2004
%F A000110 For n>0, a(n) = Aitken(n-1, n-1) [i.e. a(n-1, n-1) of Aiken's array (A011971)]
- Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 26 2004
%F A000110 Bell(n) = n + 2*C(n-2, 1) + 6*C(n-3, 1) + C(n-2, 2) + 14*C(n-4, 1) +
12*C(n-3, 2) + 30*C(n-5, 1) + 61*C(n-4, 2) + 10*C(n-3, 3) + 62*C(n-6,
1) + 240*C(n-5, 2) + 124*C(n-4, 3) + 3*C(n-3, 4) + 126*C(n-7, 1)
+ 841*C(n-6, 2) + 890*C(n-5, 3) + 131*C(n-4, 4) + 254*C(n-8, 1) +
2772*C(n-7, 2) + 5060*C(n-6, 3) + 1830*C(n-5, 4) + 70*C(n-4, 5) +
510*C(n-9, 1) + 8821*C(n-8, 2) + 25410*C(n-7, 3) + 16990*C(n-6, 4)
+ 2226*C(n-5, 5) + 15*C(n-4, 6) + 1022*C(n-10, 1) + ..... . - Andre
F. Labossiere (boronali(AT)laposte.net), Feb 22 2005
%F A000110 a(n)=sum{k=1..n, (1/k!)*sum{i=1..k, (-1)^(k-i)*binomial(k, i)*i^n}}+0^n;
- Paul Barry (pbarry(AT)wit.ie), Apr 18 2005
%F A000110 a(n) = A032347(n) + A040027(n+1) - Jon Perry (perry(AT)globalnet.co.uk),
Apr 26 2005
%F A000110 a(n) = 2*n!/(pi*e)*Im( integral_{0}^{pi} e^(e^(e^(ix))) sin(nx) dx )
where Im denotes imaginary part [Cesaro]. - David Callan (callan(AT)stat.wisc.edu),
Sep 03 2005
%F A000110 O.g.f.: A(x) = 1/(1-x-x^2/(1-2*x-2*x^2/(1-3*x-3*x^2/(1-... -n*x-n*x^2/
(1- ...))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com),
Jan 17 2006
%F A000110 Representation of Bell numbers B(n), n=1,2..., as special values of hypergeometric
function of type (n-1)F(n-1), in Maple notation: B(n)=exp(-1)*hypergeom([2,
2...2],[1,1...1],1), n=1,2..., i.e. having n-1 parameters all equal
to 2 in the numerator, having n-1 parameters all equal to 1 in the
denominator and the value of the argument equal to 1. Examples: B(1)=evalf(exp(-1)*hypergeom([],
[],1))= 1 B(2)=evalf(exp(-1)*hypergeom([2],[1],1))= 2 B(3)=evalf(exp(-1)*hypergeom([2,
2],[1,1],1))= 5 B(4)=evalf(exp(-1)*hypergeom([2,2,2],[1,1,1],1))=
15 B(5)=evalf(exp(-1)*hypergeom([2,2,2,2],[1,1,1,1],1))=52 - Karol
A. Penson (penson(AT)lptl.jussieu.fr), Jan 14 2007
%F A000110 a(n+1) = 1+sum(sum((binomial(k,i))*(2^(k-i))*(a(i)),i = 0..k),k = 0..n-1)
- Yalcin Aktar (aktaryalcin(AT)msn.com), Feb 27 2007 [There was an
error in this formula, so it should be checked carefully]
%F A000110 a(n) = [1,0,0,...,0] T^(n-1) [1,1,1,...,1]', where T is the n X n matrix
with main diagonal {1,2,3,...,n}, 1's on the diagonal immediately
above and 0's elsewhere. [Meier]
%F A000110 a(n) = ((2*n!)/(pi * e)) * ImaginaryPart(Integral[from 0 to pi](e^e^e^(i*theta))*sin(n*theta)
dtheta). - Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 27 2007
%F A000110 Formulae and comments from Tom Copeland, Oct 10 2007 (Start): a(n) =
T(n,1) = sum(j=0,...,n) S2(n,j) = sum(j=0,...,n) E(n,j) * Lag(n,-1,
j-n) = sum(j=0,...,n) [ E(n,j)/n! ] * [ n!*Lag(n,-1, j-n) ] where
T(n,x) are the Bell / Touchard / exponential polynomials; S2(n,j),
the Stirling numbers of the second kind; E(n,j), the Eulerian numbers;
and Lag(n,x,m), the associated Laguerre polynomials of order m. Note
that E(n,j)/n! = E(n,j)/ {sum(k=0,..,n) E(n,k)}.
%F A000110 The Eulerian numbers count the permutation ascents and the expression
[n!*Lag(n,-1, j-n)] is A086885 with a simple combinatorial interpretation
in terms of seating arrangements, giving a combinatorial interpretation
to n!*a(n) = sum(j=0,...,n) {E(n,j) * [n!*Lag(n,-1, j-n)]} . (End)
%F A000110 Define f_1(x),f_2(x),... such that f_1(x)=e^x and for n=2,3,... f_{n+1}(x)=diff(x*f_n(x),
x). Then for Bell numbers B_n we have B_n=1/e*f_n(1). - Milan R.
Janjic (agnus(AT)blic.net), May 30 2008
%F A000110 a(n) = (n-1)! sum_{k=1}^{n} a(n-k)/((n-k)! (k-1)!) where a(0)=1. [From
Thomas Wieder (thomas.wieder(AT)t-online.de), Sep 09 2008]
%F A000110 a(n+k) = Sum_{m=0..n} Stirling2(n,m) Sum_{r=0..k} binomial(k,r) m^r a(k-r).
- David Pasino (davepasino(AT)yahoo.com), Jan 25 2009. (Umbrally,
this may be written as a(n+k) = Sum_{m=0..n} Stirling2(n,m) (a+m)^k.
- N. J. A. Sloane (njas(AT)research.att.com), Feb 07 2009.)
%F A000110 Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009:
%F A000110 a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}
%F A000110 delta(l_1,l_2,...,l_i,...,l_n)
%F A000110 where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i > l_(i+1) and l_(i+1)
<> 0
%F A000110 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.
%e A000110 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 + 4140*x^8
+ ...
%p A000110 A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1,
i)*A000110(n-1-i),i=0..n-1); fi; end; # version 1
%p A000110 A := series(exp(exp(x)-1),x,60); A000110 := n->n!*coeff(A,x,n); # version
2
%p A000110 a:=n->sum(stirling2(n, k), k=0..n): seq(a(n), n=1..22); - Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jun 28 2007
%p A000110 a:=array(0..200); a[0]:=1; a[1]:=1; lprint(0,1); lprint(1,1); M:=200;
for n from 2 to M do a[n]:=add(binomial(n-1,i)*a[n-1-i],i=0..n-1);
lprint(n,a[n]); od:
%p A000110 with(combstruct); spec := [S, {S=Set(U,card >= 1), U=Set(Z,card >= 1)},
labeled]; [seq(combstruct[count](spec, size=n), n=0..40)];
%p A000110 G:={P=Set(Set(Atom,card>0))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,
G,labeled],size=i),i=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Dec 16 2007
%p A000110 A000110 := proc(n::integer) local k,Resultat; if n = 0 then Resultat:=1:
RETURN(Resultat); end if; Resultat:=0: for k from 1 to n do Resultat:=Resultat+A000110(n-k)/
((n-k)!*(k-1)!): od; Resultat:=Resultat*(n-1)!; RETURN(Resultat);
end proc; [From Thomas Wieder (thomas.wieder(AT)t-online.de), Sep
09 2008]
%t A000110 f[n_] := Sum[ StirlingS2[n, k], {k, 1, n}]; Table[ f[n], {n, 0, 21}]
(from Robert G. Wilson v)
%o A000110 (PARI) a(n)=local(m); if(n<0,0,m=contfracpnqn(matrix(2,n\2,i,k,if(i==1,
-k*x^2,1-(k+1)*x))); polcoeff(1/(1-x+m[2,1]/m[1,1])+x*O(x^n),n))
(from Michael Somos)
%o A000110 (PARI) a(n)=polcoeff(sum(k=0,n,prod(i=1,k,x/(1-i*x)),x^n*O(x)),n) /*
Michael Somos, Aug 22 2004 */
%o A000110 (PARI) a(n)=exp(-1)*suminf(k=0,1.0*k^(n-1)/k!) - Gottfried Helms, Mar
30 2007
%o A000110 (PARI) m=matpascal(5)-matid(6); pe=matid(6)+m/1! + m^2/2!+m^3/3!+m^4/
4!+m^5/5! ; a(n) = pe[n,1] - Gottfried Helms, Apr 10 2007
%o A000110 sage: from sage.combinat.expnums import expnums2 sage: expnums2(30, 1)
- Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008
%o A000110 (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) -
1), n))} /* Michael Somos, Jun 28 2009 */
%o A000110 (PARI) a(n) = polcoeff( exp(exp(x)) ,n-1) / polcoeff( exp(x+1) ,n-1)
[From Gottfried Helms (helms(AT)uni-kassel.de), Jul 25 2009]
%Y A000110 Partial sums give A005001. Cf. A049020. See A061462 for powers of 2 dividing
a(n).
%Y A000110 Right-most diagonal of triangle A121207.
%Y A000110 Cf. A000311, A103293, A084423.
%Y A000110 Cf. A005001, A087650, A029761, A024716, A000296, A058692, A060719.
%Y A000110 Cf. A008277, A000166, A000255, A000108, A000045, A000204.
%Y A000110 Cf. A094262, A008277, A005001, A003422, A000166, A000204, A000045, A000108.
%Y A000110 a(n) = A123158(n, 0).
%Y A000110 A144293 gives largest prime factor.
%Y A000110 Equals row sums of triangle A152432 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Dec 04 2008]
%Y A000110 A165194 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2009]
%Y A000110 A165196 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2009]
%Y A000110 Sequence in context: A099263 A164863 A164864 this_sequence A134381 A107589
A006790
%Y A000110 Adjacent sequences: A000107 A000108 A000109 this_sequence A000111 A000112
A000113
%K A000110 core,nonn,easy,nice
%O A000110 0,3
%A A000110 N. J. A. Sloane (njas(AT)research.att.com).
%E A000110 Replaced arXiv URL by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Oct 23 2009
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