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What are the Bell Numbers?

You come across a reference to a special number sequence, perhaps the Catalan numbers, or the Fibonacci numbers, the Motzkin numbers, the Bell numbers, the lucky numbers, the tetrahedral numbers, the abundant or deficient numbers, etc., and you would like to find out what they are.

Let us take the Bell numbers as an example.

Again you begin by going to the OEIS, where at the foot of the page you see a link to the Index.

You click on the Index, and go to section Be, where you find something like the following.


Beethoven: A001491, A054245, A123456
Beethoven: see also music
beginning with t: A006092, A005224
Belgian numbers: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043, A107062, A107070.
Bell numbers, sequences related to :

Bell numbers: A000110*
Bell numbers: see also A007311
Bell numbers: see also set partitions
Bell numbers: see also Stirling numbers of 2nd kind

bell ringing , sequences related to

bell ringing: (1) A090277 A090278 A090279 A090280 A090281 A090282 A090283 A090284
bell ringing: (2) A057112 A060112 A060135

(The asterisks indicate the most important sequences.)

Clicking on A000110 produces the following reply, which is just what you wanted. The entry gives the beginning of the sequence, a description, references, formulae, Maple and Mathematica programs to generate the sequence, links, cross-references, etc.

This is one of the longest and most interesting entries in the database.

Greetings from the On-Line Encyclopedia of Integer Sequences!

A000110 Bell or exponential numbers: ways of placing n labeled balls into n indistinguishable boxes.
(Formerly M1484 N0585)
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372, 474869816156751, 4506715738447323, 44152005855084346 (list; graph; refs; listen; history; edit; internal format)



Number of partitions of a set of n labeled elements.

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

If an integer is squarefree 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

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

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

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

Comment from Neven Juric, Oct 19 2009: (Start)

The case n=4: aaaa, aaab, aaba, aabb, aabc, abaa, abab, abac, abba, abbb, abbc, abca, abcb, abcc, abcd

The case n=5:

aaaaa, aaaab, aaaba, aaabb, aaabc, aabaa, aabab, aabac, aabba, aabbb,

aabbc, aabca, aabcb, aabcc, aabcd, abaaa, abaab, abaac, ababa, ababb,

ababc, abaca, abacb, abacc, abacd, abbaa, abbab, abbac, abbba, abbbb,

abbbc, abbca, abbcb, abbcc, abbcd, abcaa, abcab, abcac, abcad, abcba,

abcbb, abcbc, abcbd, abcca, abccb, abccc, abccd, abcda, abcdb, abcdc,

abcdd, abcde (End)

In other words, number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=1+max(prefix) for k>=1, see example (cf. A080337 and A189845). [Joerg Arndt, Apr 30 2011]

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

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

Triangle (addition) scheme to produce terms, derived from the recurrence, from Oscar Arevalo (loarevalo(AT)sbcglobal.net), May 11 2005:


1 2

2 3 5

5 7 10 15

15 20 27 37 ... [This is Aitken's array A011971]

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

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

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

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

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.

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" reference 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

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

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) =









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:





...15.....5....2....1...1.... (X) P



..877...203...52...15...5...2 (End)

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.

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.

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

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

This is the exp transform of A000012. [From Thomas Wieder (thomas.wieder(AT)t-online.de), Sep 09 2008]

Contribution from A. Umar (aumarh(AT)squ.edu.om), Oct 12 2008: (Start)

a(n) is also the number of idempotent order-decreasing full transformations (of an n-chain).

a(n) is also the number of nilpotent partial one-one order-decreasing transformations (of an n-chain).

a(n+1) is also the number of partial one-one order-decreasing transformations (of an n-chain). (End)

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 19 2008: (Start)

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).

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.


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]

Contribution from David Pasino (davpas(AT)charter.net), Dec 04 2008: (Start)

The entries f(i, j) in the exponential of the infinite lower-triangular matrix

of binomial coefficients b(i, j) are f(i, j) = b(i, j) e a(i - j). (End)

Equals Lim_{k->inf.} A071919^k [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 02 2009]

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]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 14 2009: (Start)

Repeated iterates of (binomial transform of [1,0,0,0,...]) will converge upon

(1, 2, 5, 15, 52,...) when each result is prefaced with a "1"; such that the

final result is the fixed limit: (binomial transform of [1,1,2,5,15,...] = (1,2,5,15,52,...). (End)

Contribution from Karol A. Penson (penson(AT)lptl.jussieu.fr), May 03 2009: (Start)

Relation between the Bell numbers B(n) and the n-th derivative of 1/Gamma(1+x) of such derivatives through seq(subs(x=0, simplify(diff(GAMMA(1+x)^(-1),x$n))), n=1..6);

b) leave them expressed in terms of digamma (Psi(k)) and polygamma (Psi(k,n)) functions and unevaluated ;

Examples of such expressions, for n=1..5, are :

n=1: -Psi(1),

n=2: -(-Psi(1)^2+Psi(1,1)),

n=3: -Psi(1)^3+3*Psi(1)*Psi(1,1)-Psi(2,1),

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)),

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)

+10*Psi(1,1)*Psi(2,1)+5*Psi(1)*Psi(3,1)-Psi(4,1) ;

c) for a given n read off the sum of absolute values of coefficients of every term involving digamma or polygamma functions.

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;

d) Observe that this decomposition of the Bell number B(n) apparently does not involve the Stirling numbers of the second kind explicitly. (End)

The numbers given above by Penson lead to the multinomial coefficients A036040. - Johannes W. Meijer, Aug 14 2009

Column 1 of A162663. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jul 09 2009]

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

Starting with offset 1 = row sums of triangle A165194 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2009: (Start)

a(n+1) = A165196(2^n); where A165196 begins: (1, 2, 4, 5, 7, 12, 14, 15,...).

such that A165196(2^3) = 15 = A000110(4). (End)

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

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.


a(n)= E(X^n), i.e. the n-th moment about the origin of a random variable X that has a Poisson distribution with (rate) parameter, lambda = 1. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Nov 30 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 09 2010: (Start)

Let A000110 = S(x), then S(x) = A(x)/A(x^2) when A(x) = A173110;

or (1, 1, 2, 5, 15, 52,...) = (1, 1, 3, 6, 20, 60,...) / (1, 0, 1, 0, 3, 0, 6, 0, 20,...). (End)

The Bell numbers serve as the upper limit for the number of unique homomorphic images from any given finite universal algebra. Every algebra homomorphism is determined by its kernel, which must be a congruence relation. As the number of possible congruence relations with respect to a finite universal algebra must be a subset of its possible equivalence classes (given by the bell numbers), it follows naturally. [From Max Sills (sillsm(AT)gmail.com), Jun 01 2010]

For a proof of the o.g.f. given in the R. Stephan comment see e.g. the W. Lang link under A071919. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 23 2010]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 08 2010: (Start)

Let B(x) = (1 + x + 2x^2 + 5x^3 + ...). Then B(x) is satisfied by A(x)/A(x^2)

where A(x) = polcoeff A173110: (1 + x + 3x^2 + 6x^3 + 20x^4 + 60x^5 + ...) =

B(x) * B(x^2) * B(x^4) * B(x^8) * ... (End)

Consider a set of A000217(n) balls of n colors in which, for each integer k = 1 to n, exactly one color appears in the set a total of k times. (Each ball has exactly one color and is indistinguishable from other balls of the same color.) a(n+1) equals the number of ways to choose 0 or more balls of each color without choosing any two colors the same positive number of times.  (See related comments for A000108, A008277, A016098.) [From Matthew Vandermast(ghodges14(AT)comcast.net), Nov 22 2010]

A binary counter with faulty bits starts at value 0 and attempts to increment by 1 at each step. Each bit that should toggle may or may not do so. a(n) is the number of ways that the counter can have the value 0 after n steps. e.g. n=3, the 5 trajectories are 0,0,0,0; 0,1,0,0; 0,1,1,0; 0,0,1,0; 0,1,3,0 - [from David Scambler (dscambler(AT)bmm.com) Jan 24 2011]

No Bell number is divisible by 8, see the table 1.7 by Lunnon, Pleasants and Stephens. [From Jon Perry, 07 Feb 2011]

a(n+1) is the number of (symmetric) positive semidefinite n x n 0-1 matrices.  These correspond to equivalence relations on {1,...,n+1}, where matrix element M[i,j] = 1 if and only if i and j are equivalent to each other but not to n+1. [from Robert Israel (israel(AT)math.ubc.ca) Mar 16 2011]


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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge; see Section 1.4 and Example 5.2.4.

Umar, A. On the semigroups of order-decreasing finite full transformations, Proc. Roy. Soc. Edinburgh 120A (1992), 129-142. [From A. Umar (aumarh(AT)squ.edu.om), Oct 12 2008]

Umar, A. On the semigroups of partial one-to-one order-decreasing finite transformations, Proc. Roy. Soc. Edinburgh 123A (1993), 355-363. [From A. Umar (aumarh(AT)squ.edu.om), Oct 12 2008]


S. Plouffe, Table of n, a(n) for n = 0..500

Joerg Arndt, Fxtbook, p.151, p.358, and p.368

R. Aldrovandi and L. P. Freitas, Continuous iteration of dynamical maps

Pat Ballew, Bell Numbers

D. Barsky, Analyse p-adique et suites classiques de nombres, Sem. Loth. Comb. B05b (1981) 1-21.

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

H. Bottomley, Illustration of initial terms

A. Burstein and I. Lankham, Combinatorics of patience sorting piles

David Callan, A Combinatorial Interpretation of the Eigensequence for Composition.

David Callan, Cesaro's Integral Formula for the Bell Numbers (Corrected).

David Callan, Cesaro's integral formula for the Bell numbers (corrected).

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

A. Claesson and T. Mansour, Counting patterns of type (1,2) or (2,1).

C. Cooper and R. E. Kennedy, Patterns, automata and Stirling numbers of the second kind, Mathematics and Computer Education Journal, 26 (1992), 120-124. [From Peter Bala (pbala(AT)toucansurf.com), Oct 19 2008]

R. M. Dickau, Bell number diagrams

R. Ehrenborg and M. Readdy, Juggling and applications to q-analogues, Discrete Math. 157 (1996), 107-125.

John Fiorillo, GENJI-MON

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 109, 110

H. Fripertinger, The Bell Numbers

Daniel L. Geisler, Combinatorics of Iterated Functions

A. Gertsch and A. M.Robert, Some congruences concerning the Bell numbers

Gottfried Helms, Bell Numbers, 2008.

A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 15

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 65

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 73

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 291

A. Knutson, Siteswap FAQ, Section 5, Working backwards, defines the term "orbit" in siteswap notation.

Kazuhiro Kunii, Genji-koh no zu [Japanese page illustrating a(5) = 52]

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Peter Luschny, Set partitions and Bell numbers

R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets

A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.

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. (pdf, ps)

K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem

K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)

K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, 4 (2001), #01.2.5.

G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

S. Plouffe, Table of n, a(n) for n = 0..3015

S. Plouffe, Bell numbers (first 1000 terms)

T. Prellberg, On the asymptotic analysis of a class of linear recurrences (slides).

C. Radoux, Determinants de Hankel et theoreme de Sylvester

S. Ramanujan, Notebook entry

A. Ross, PlanetMath.org, Bell number

T. Sillke, Bell numbers

A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, Combinatorial physics, normal order and model Feynman graphs.

A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, Partition functions and graphs: A combinatorial approach.

Michael Z. Spivey, A generalized recurrence for Bell numbers, J. Integer Sequences, Vol. 11 (2008), Article 08.2.5. [From Michael Z. Spivey (mspivey(AT)pugetsound.edu), May 26 2010]

Eric Weisstein's World of Mathematics, Bell Number

Eric Weisstein's World of Mathematics, Binomial Transform

Eric Weisstein's World of Mathematics, Stirling Transform

Eric Weisstein's World of Mathematics, Bell Triangle

H. S. Wilf, Generatingfunctionology</a

>, 2nd edn., Academic Press, NY, 1994, pp. 21ff.

Index entries for sequences related to rooted trees

Index entries for "core" sequences

Index entries for sequences related to juggling

Index entries for related partition-counting sequences


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.

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

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.

a(n) = exp(-1)*sum(k=>0, k^n/k!) [Dobinski] - Benoit Cloitre (benoit7848c(AT)orange.fr), May 19 2002

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.

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

G.f.: sum{k>=0, x^k/prod[l=1..k, 1-lx]}. - R. Stephan, Apr 18 2004

a(n+1) = exp(-1)*sum(k=>0, (k+1)^(n)/k!) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 03 2004

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

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

a(n) = A032347(n) + A040027(n+1) - Jon Perry (perry(AT)globalnet.co.uk), Apr 26 2005

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

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

Contribution by Karol A. Penson (penson(AT)lptl.jussieu.fr), Jan 14 2007 (Start)

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.


B(1)=exp(-1)*hypergeom([],[],1)= 1

B(2)=exp(-1)*hypergeom([2],[1],1)= 2

B(3)=exp(-1)*hypergeom([2,2],[1,1],1)= 5

B(4)=exp(-1)*hypergeom([2,2,2],[1,1,1],1)= 15


(Warning: this formula is correct but its application by a computer may not yield exact results, especially with large number of parameters)


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]

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]

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

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)}.

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)

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

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]

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.)

Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009:

a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}


where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i > l_(i+1) and l_(i+1) <> 0

and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.

Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]:=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). [From Milan R. Janjic (agnus(AT)blic.net), Jul 08 2010]


1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 + 4140*x^8 + ...

Restricted growth strings (RGS): For n=0 there is one empty string; for n=1 there is one string [0]; for n=2 there are 2 strings [00], [01]; for n=3 there are a(3)=5 strings [000], [001], [010], [011], and [012]; for n=4 there are a(4)=15 strings

1:  [0000],

2:  [0001],

3:  [0010],

4:  [0011],

5:  [0012],

6:  [0100],

7:  [0101],

8:  [0102],

9:  [0110],

10:  [0111],

11:  [0112],

12:  [0120],

13:  [0121],

14:  [0122],

15:  [0123]. [Joerg Arndt, Apr 30 2011]


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

A := series(exp(exp(x)-1), x, 60); A000110 := n->n!*coeff(A, x, n); # version 2

a:=n->sum(stirling2(n, k), k=0..n): seq(a(n), n=1..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007

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:

with(combstruct); spec := [S, {S=Set(U, card >= 1), U=Set(Z, card >= 1)}, labeled]; [seq(combstruct[count](spec, size=n), n=0..40)];

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

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]

A000110 := n -> combinat[bell](n): # - Peter Luschny, Mar 30, 2011


f[n_] := Sum[ StirlingS2[n, k], {k, 1, n}]; Table[ f[n], {n, 0, 21}] (from Robert G. Wilson v)

Table[BellB[n], {n, 200}]  (* From Harvey P. Dale, Mar 1 2011 *)


(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)

(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 */

(PARI) a(n)=exp(-1)*suminf(k=0, 1.0*k^(n-1)/k!) - Gottfried Helms, Mar 30 2007

(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

(Sage) from sage.combinat.expnums import expnums2;

expnums2(30, 1) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1), n))} /* Michael Somos, Jun 28 2009 */

(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]

# (Python) The objective of this implementation is efficiency.

# m -> [a(0), a(1), ..., a(m)] for m > 0.

def A000110_list(m):

....A = [0 for i in range(0, m)]

....A[0] = 1

....R = [1, 1]

....for n in range(1, m):

........A[n] = A[0]

........for k in range(n, 0, -1):

............A[k-1] += A[k]


....return R

A000110_list(100) # example call - Peter Luschny, Jan 18 2011

(MAGMA) [Bell(n): n in [0..80]]; // From Vincenzo Librandi, Feb 07 2011

(Maxima) makelist(belln(n), n, 0, 80); [Emanuele Munarini, Jul 04 2011]


Partial sums give A005001. See A061462 for powers of 2 dividing a(n).

Right-most diagonal of triangle A121207. A144293 gives largest prime factor.  a(n) = A123158(n, 0).

Cf. A049020, A000311, A103293, A084423, A005001, A087650, A029761, A024716, A000296, A058692, A060719, A008277, A000166, A000255, A000108, A000045, A000204, A094262, A008277, A005001, A003422, A000166, A000204, A000045, A000108.

Equals row sums of triangle A152432. Cf. A165194, A165196, A173110, A173110. [From Gary W. Adamson (qntmpkt(AT)yahoo.com)]

Sequence in context: A192127 A192867 A192128 * A186001 A134381 A107589

Adjacent sequences:  A000107 A000108 A000109 * A000111 A000112 A000113




N. J. A. Sloane (njas(AT)research.att.com).


Replaced arXiv URL by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2009

Typo in link corrected by Matthew Vandermast (ghodges14(AT)comcast.net), Nov 27 2010

Thousands of other important sequences can also be located through the Index to the database.

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