This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000110 M1484 N0585 #1379 Jun 02 2025 16:49:53
%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 5832742205057,51724158235372,474869816156751,4506715738447323,44152005855084346,445958869294805289,4638590332229999353,49631246523618756274
%N A000110 Bell or exponential numbers: number of ways to partition a set of n labeled elements.
%C A000110 The leading diagonal of its difference table is the sequence shifted, see Bernstein and Sloane (1995). - _N. J. A. Sloane_, Jul 04 2015
%C A000110 Also the number of equivalence relations that can be defined on a set of n elements. - Federico Arboleda (federico.arboleda(AT)gmail.com), Mar 09 2005
%C A000110 a(n) = number of nonisomorphic colorings of a map consisting of a row of n+1 adjacent regions. Adjacent regions cannot have the same color. - _David W. Wilson_, Feb 22 2005
%C A000110 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_, 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_, 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], [1,2], [1,1] = 15; etc. - _Jon Perry_, Mar 05 2004
%C A000110 Number of distinct rhyme schemes for a poem of n lines: a rhyme scheme is a string of letters (e.g., '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; also see example by Neven Juric. - _Bill Blewett_, Mar 23 2004
%C A000110 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
%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. - _Augustine O. Munagi_, 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 52
%C A000110 ... [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, one has: a(n) = Sum_{i=1..P(n)} (n!/(Product_{j=1..p(i)} p(i,j)!)) * (1/(Product_{j=1..d(i)} m(i,j)!)). - _Thomas Wieder_, May 18 2005
%C A000110 a(n+1) is 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 the rule from Jon Perry, Mar 05 2004, is used, 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_, 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_, May 01 2006
%C A000110 a(n) is the 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_, Oct 07 2006
%C A000110 From _Gottfried Helms_, Mar 30 2007: (Start)
%C A000110 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 1
%C A000110 877 1421 1092 525 175 42 7 1
%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 1
%C A000110 877 203 52 15 5 2 1 1
%C A000110 (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_, Apr 10 2007
%C A000110 When n is prime, a(n) == 2 (mod n), but the converse is not always true. 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. - _David W. Wilson_, Aug 04 2007 and Sep 24 2007
%C A000110 This sequence and A000587 form a reciprocal pair under the list partition transform described in A133314. - _Tom Copeland_, 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_, Jan 21 2008
%C A000110 This is the exponential transform of A000012. - _Thomas Wieder_, Sep 09 2008
%C A000110 From _Abdullahi Umar_, 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 From _Peter Bala_, 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. - _Gary W. Adamson_, Dec 04 2008
%C A000110 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). - _David Pasino_, Dec 04 2008
%C A000110 Equals lim_{k->oo} A071919^k. - _Gary W. Adamson_, 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. - _Gary W. Adamson_, Jan 04 2009
%C A000110 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,...). - _Gary W. Adamson_, Jan 14 2009
%C A000110 From _Karol A. Penson_, May 03 2009: (Start)
%C A000110 Relation between the Bell numbers B(n) and the n-th derivative of 1/Gamma(1+x) evaluated at x=1:
%C A000110 a) produce a number of such derivatives through seq(subs(x=1, simplify((d^n/dx^n)GAMMA(1+x)^(-1))), n=1..5);
%C A000110 b) leave them expressed in terms of digamma (Psi(k)) and polygamma (Psi(k,n)) functions and unevaluated;
%C A000110 Examples of such expressions, for n=1..5, are:
%C A000110 n=1: -Psi(1),
%C A000110 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) + 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 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.
%C A000110 (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. - _Franklin T. Adams-Watters_, 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. - _Gary W. Adamson_, Sep 06 2009
%C A000110 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). - _Gary W. Adamson_, Sep 06 2009
%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. - _Johannes W. Meijer_, Oct 16 2009
%C A000110 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. - _Geoffrey Critzer_, Nov 30 2009
%C A000110 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, ...). - _Gary W. Adamson_, Feb 09 2010
%C A000110 The Bell numbers serve as the upper limit for the number of distinct 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. - _Max Sills_, Jun 01 2010
%C A000110 For a proof of the o.g.f. given in the R. Stephan comment see, e.g., the W. Lang link under A071919. - _Wolfdieter Lang_, Jun 23 2010
%C A000110 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) * .... - _Gary W. Adamson_, Jul 08 2010
%C A000110 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.) - _Matthew Vandermast_, Nov 22 2010
%C A000110 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., for 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. - _David Scambler_, Jan 24 2011
%C A000110 No Bell number is divisible by 8, and no Bell number is congruent to 6 modulo 8; see Theorem 6.4 and Table 1.7 in Lunnon, Pleasants and Stephens. - _Jon Perry_, Feb 07 2011, clarified by _Eric Rowland_, Mar 26 2014
%C A000110 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. - _Robert Israel_, Mar 16 2011
%C A000110 a(n) is the number of monotonic-labeled forests on n vertices with rooted trees of height less than 2. We note that a labeled rooted tree is monotonic-labeled if the label of any parent vertex is greater than the label of any offspring vertex. See link "Counting forests with Stirling and Bell numbers". - _Dennis P. Walsh_, Nov 11 2011
%C A000110 a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000772 and A094198. - _Peter Bala_, Nov 25 2011
%C A000110 B(n) counts the length n+1 rhyme schemes without repetitions. E.g., for n=2 there are 5 rhyme schemes of length 3 (aaa, aab, aba, abb, abc), and the 2 without repetitions are aba, abc. This is basically O. Munagi's result that the Bell numbers count partitions into subsets of nonconsecutive integers (see comment above dated Mar 20 2005). - Eric Bach, Jan 13 2012
%C A000110 Right and left borders and row sums of A212431 = A000110 or a shifted variant. - _Gary W. Adamson_, Jun 21 2012
%C A000110 Number of maps f: [n] -> [n] where f(x) <= x and f(f(x)) = f(x) (projections). - _Joerg Arndt_, Jan 04 2013
%C A000110 Permutations of [n] avoiding any given one of the 8 dashed patterns in the equivalence classes (i) 1-23, 3-21, 12-3, 32-1, and (ii) 1-32, 3-12, 21-3, 23-1. (See Claesson 2001 reference.) - _David Callan_, Oct 03 2013
%C A000110 Conjecture: No a(n) has the form x^m with m > 1 and x > 1. - _Zhi-Wei Sun_, Dec 02 2013
%C A000110 Sum_{n>=0} a(n)/n! = e^(e-1) = 5.57494152476..., see A234473. - _Richard R. Forberg_, Dec 26 2013 (This is the e.g.f. for x=1. - _Wolfdieter Lang_, Feb 02 2015)
%C A000110 Sum_{j=0..n} binomial(n,j)*a(j) = (1/e)*Sum_{k>=0} (k+1)^n/k! = (1/e) Sum_{k=1..oo} k^(n+1)/k! = a(n+1), n >= 0, using the Dobinski formula. See the comment by _Gary W. Adamson_, Dec 04 2008 on the Pascal eigensequence. - _Wolfdieter Lang_, Feb 02 2015
%C A000110 In fact it is not really an eigensequence of the Pascal matrix; rather the Pascal matrix acts on the sequence as a shift. It is an eigensequence (the unique eigensequence with eigenvalue 1) of the matrix derived from the Pascal matrix by adding at the top the row [1, 0, 0, 0 ...]. The binomial sum formula may be derived from the definition in terms of partitions: label any element X of a set S of N elements, and let X(k) be the number of subsets of S containing X with k elements. Since each subset has a unique coset, the number of partitions p(N) of S is given by p(N) = Sum_{k=1..N} (X(k) p(N-k)); trivially X(k) = N-1 choose k-1. - _Mason Bogue_, Mar 20 2015
%C A000110 a(n) is the number of ways to nest n matryoshkas (Russian nesting dolls): we may identify {1, 2, ..., n} with dolls of ascending sizes and the sets of a set partition with stacks of dolls. - _Carlo Sanna_, Oct 17 2015
%C A000110 Number of permutations of [n] where the initial elements of consecutive runs of increasing elements are in decreasing order. a(4) = 15: `1234, `2`134, `23`14, `234`1, `24`13, `3`124, `3`2`14, `3`24`1, `34`12, `34`2`1, `4`123, `4`2`13, `4`23`1, `4`3`12, `4`3`2`1. - _Alois P. Heinz_, Apr 27 2016
%C A000110 Taking with alternating signs, the Bell numbers are the coefficients in the asymptotic expansion (Ramanujan): (-1)^n*(A000166(n) - n!/exp(1)) ~ 1/n - 2/n^2 + 5/n^3 - 15/n^4 + 52/n^5 - 203/n^6 + O(1/n^7). - _Vladimir Reshetnikov_, Nov 10 2016
%C A000110 Number of treeshelves avoiding pattern T231. See A278677 for definitions and examples. - _Sergey Kirgizov_, Dec 24 2016
%C A000110 Presumably this satisfies Benford's law, although the results in Hürlimann (2009) do not make this clear. - _N. J. A. Sloane_, Feb 09 2017
%C A000110 a(n) = Sum(# of standard immaculate tableaux of shape m, m is a composition of n), where this sum is over all integer compositions m of n > 0. This formula is easily seen to hold by identifying standard immaculate tableaux of size n with set partitions of { 1, 2, ..., n }. For example, if we sum over integer compositions of 4 lexicographically, we see that 1+1+2+1+3+3+3+1 = 15 = A000110(4). - _John M. Campbell_, Jul 17 2017
%C A000110 a(n) is also the number of independent vertex sets (and vertex covers) in the (n-1)-triangular honeycomb bishop graph. - _Eric W. Weisstein_, Aug 10 2017
%C A000110 Even-numbered entries represent the numbers of configurations of identity and non-identity for alleles of a gene in n diploid individuals with distinguishable maternal and paternal alleles. - _Noah A Rosenberg_, Jan 28 2019
%C A000110 Number of partial equivalence relations (PERs) on a set with n elements (offset=1), i.e., number of symmetric, transitive (not necessarily reflexive) relations. The idea is to add a dummy element D to the set, and then take equivalence relations on the result; anything equivalent to D is then removed for the partial equivalence relation. - _David Spivak_, Feb 06 2019
%C A000110 Number of words of length n+1 with no repeated letters, when letters are unlabeled. - _Thomas Anton_, Mar 14 2019
%C A000110 Named by Becker and Riordan (1948) after the Scottish-American mathematician and writer Eric Temple Bell (1883 - 1960). - _Amiram Eldar_, Dec 04 2020
%C A000110 Also the number of partitions of {1,2,...,n+1} with at most one n+1 singleton. E.g., a(3)=5: {13|24, 12|34, 123|4, 14|23, 1234}. - _Yuchun Ji_, Dec 21 2020
%C A000110 a(n) is the number of sigma algebras on the set of n elements. Note that each sigma algebra is generated by a partition of the set. For example, the sigma algebra generated by the partition {{1}, {2}, {3,4}} is {{}, {1}, {2}, {1,2}, {3,4}, {1,3,4}, {2,3,4}, {1,2,3,4}}. - _Jianing Song_, Apr 01 2021
%C A000110 a(n) is the number of P_3-free graphs on n labeled nodes. - _M. Eren Kesim_, Jun 04 2021
%C A000110 a(n) is the number of functions X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple abc we have X(ab)X(ac)X(bc) not in {+-+,++-,-++}. - _Robert Lauff_, Dec 09 2022
%C A000110 From _Manfred Boergens_, Mar 11 2025: (Start)
%C A000110 The partitions in the definition can be described as disjoint covers of the set. "Covers" in general give rise to the following amendments:
%C A000110 For disjoint covers which may include one empty set see A186021.
%C A000110 For arbitrary (including non-disjoint) covers see A003465.
%C A000110 For arbitrary (including non-disjoint) covers which may include one empty set see A000371. (End)
%D A000110 Stefano Aguzzoli, Brunella Gerla and Corrado Manara, Poset Representation for Goedel and Nilpotent Minimum Logics, in Symbolic and Quantitative Approaches to Reasoning with Uncertainty, Lecture Notes in Computer Science, Volume 3571/2005, Springer-Verlag. [Added by _N. J. A. Sloane_, Jul 08 2009]
%D A000110 S. Ainley, Problem 19, QARCH, No. IV, Nov 03, 1980.
%D A000110 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 205.
%D A000110 W. Asakly, A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour, S. Wagner, Set partition asymptotics and a conjecture of Gould and Quaintance, Journal of Mathematical Analysis and Applications, Volume 416, Issue 2, Aug 15 2014, Pages 672-682.
%D A000110 J. Balogh, B. Bollobas and D. Weinreich, A jump to the Bell numbers for hereditary graph properties, J. Combin. Theory Ser. B 95 (2005), no. 1, 29-48.
%D A000110 R. E. Beard, On the coefficients in the expansion of exp(exp(t)) and exp(-exp(t)), J. Institute Actuaries, 76 (1951), 152-163.
%D A000110 H. W. Becker, Abstracts of two papers related to Bell numbers, Bull. Amer. Math. Soc., 52 (1946), p. 415.
%D A000110 E. T. Bell, The iterated exponential numbers, Ann. Math., 39 (1938), 539-557.
%D A000110 C. M. Bender, D. C. Brody and B. K. Meister, Quantum Field Theory of Partitions, J. Math. Phys., 40,7 (1999), 3239-45.
%D A000110 E. A. Bender and J. R. Goldman, Enumerative uses of generating functions, Indiana Univ. Math. J., 20 (1971), 753-765.
%D A000110 G. Birkhoff, Lattice Theory, Amer. Math. Soc., Revised Ed., 1961, p. 108, Ex. 1.
%D A000110 M. T. L. Bizley, On the coefficients in the expansion of exp(lambda exp(t)), J. Inst. Actuaries, 77 (1952), p. 122.
%D A000110 J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 41.
%D A000110 Carlier, Jacques; and Lucet, Corinne; A decomposition algorithm for network reliability evaluation. In First International Colloquium on Graphs and Optimization (GOI), 1992 (Grimentz). Discrete Appl. Math. 65 (1996), 141-156.
%D A000110 Anders Claesson, Generalized Pattern Avoidance, European Journal of Combinatorics, 22 (2001) 961-971.
%D A000110 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 210.
%D A000110 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 92-93.
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%H A000110 Karen Yeats, <a href="https://arxiv.org/abs/1805.11735">A study on prefixes of c_2 invariants</a>, arXiv:1805.11735 [math.CO], 2018.
%H A000110 Alexander Yong, <a href="http://arxiv.org/abs/1309.5883">The Joseph Greenberg problem: combinatorics and comparative linguistics</a>, arXiv preprint arXiv:1309.5883 [math.CO], 2013.
%H A000110 Abdelmoumène Zekiri, Farid Bencherif, and Rachid Boumahdi, <a href="https://www.emis.de/journals/JIS/VOL21/Zekiri/zekiri4.html">Generalization of an Identity of Apostol</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.
%H A000110 Xiqiang Zhao, Shuangshuang Ding, Tingming Wang, <a href="https://doi.org/10.1016/j.disc.2003.08.007">Some summation rules related to the Riordan arrays</a>, Disc. Math. 281 (2004) 295-307
%H A000110 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000110 <a href="/index/J#Juggling">Index entries for sequences related to juggling</a>
%H A000110 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A000110 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A000110 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F A000110 E.g.f.: exp(exp(x) - 1).
%F A000110 Recurrence: a(n+1) = Sum_{k=0..n} a(k)*binomial(n, k).
%F A000110 a(n) = Sum_{k=0..n} Stirling2(n, k).
%F A000110 a(n) = Sum_{j=0..n-1} (1/(n-1)!)*A000166(j)*binomial(n-1, j)*(n-j)^(n-1). - _André F. Labossière_, Dec 01 2004
%F A000110 G.f.: (Sum_{k>=0} 1/((1-k*x)*k!))/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_, Mar 25 2002
%F A000110 a(n) = exp(-1)*Sum_{k >= 0} k^n/k! [Dobinski]. - _Benoit Cloitre_, 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(b/(b+n)) where b satisfies b*log(b) = n - 1/2 (see Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed., p. 493). - _Benoit Cloitre_, Oct 23 2002, corrected by _Vaclav Kotesovec_, Jan 06 2013
%F A000110 Lovasz (Combinatorial Problems and Exercises, North-Holland, 1993, Section 1.14, Problem 9) gives another asymptotic formula, quoted by Mezo and Baricz. - _N. J. A. Sloane_, Mar 26 2015
%F A000110 G.f.: Sum_{k>=0} x^k/(Product_{j=1..k} (1-j*x)) (see Klazar for a proof). - _Ralf Stephan_, Apr 18 2004
%F A000110 a(n+1) = exp(-1)*Sum_{k>=0} (k+1)^(n)/k!. - _Gerald McGarvey_, Jun 03 2004
%F A000110 For n>0, a(n) = Aitken(n-1, n-1) [i.e., a(n-1, n-1) of Aitken's array (A011971)]. - _Gerald McGarvey_, Jun 26 2004
%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_, Apr 18 2005
%F A000110 a(n) = A032347(n) + A040027(n+1). - _Jon Perry_, Apr 26 2005
%F A000110 a(n) = (2*n!/(Pi*e))*Im( Integral_{x=0..Pi} e^(e^(e^(ix))) sin(nx) dx ) where Im denotes imaginary part [Cesaro]. - _David Callan_, Sep 03 2005
%F A000110 O.g.f.: 1/(1-x-x^2/(1-2*x-2*x^2/(1-3*x-3*x^2/(.../(1-n*x-n*x^2/(...)))))) (continued fraction due to Ph. Flajolet). - _Paul D. Hanna_, Jan 17 2006
%F A000110 From _Karol A. Penson_, Jan 14 2007: (Start)
%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.
%F A000110 Examples:
%F A000110 B(1)=exp(-1)*hypergeom([],[],1)=1
%F A000110 B(2)=exp(-1)*hypergeom([2],[1],1)=2
%F A000110 B(3)=exp(-1)*hypergeom([2,2],[1,1],1)=5
%F A000110 B(4)=exp(-1)*hypergeom([2,2,2],[1,1,1],1)=15
%F A000110 B(5)=exp(-1)*hypergeom([2,2,2,2],[1,1,1,1],1)=52
%F A000110 (Warning: this formula is correct but its application by a computer may not yield exact results, especially with a large number of parameters.)
%F A000110 (End)
%F A000110 a(n+1) = 1 + Sum_{k=0..n-1} Sum_{i=0..k} binomial(k,i)*(2^(k-i))*a(i). - _Yalcin Aktar_, Feb 27 2007
%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_, Aug 27 2007
%F A000110 From _Tom Copeland_, Oct 10 2007: (Start)
%F A000110 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)].
%F A000110 (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) = (d/dx)(x*f_n(x)). Then for Bell numbers B_n we have B_n=1/e*f_n(1). - _Milan Janjic_, May 30 2008
%F A000110 a(n) = (n-1)! Sum_{k=1..n} a(n-k)/((n-k)! (k-1)!) where a(0)=1. - _Thomas Wieder_, 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_, Feb 07 2009)
%F A000110 Sum_{k=1..n-1} a(n)*binomial(n,k) = Sum_{j=1..n}(j+1)*Stirling2(n,j+1). - [Zhao] - _R. J. Mathar_, Jun 24 2024
%F A000110 From _Thomas Wieder_, Feb 25 2009: (Start)
%F A000110 a(n) = Sum_{k_1=0..n+1} Sum_{k_2=0..n}...Sum_{k_i=0..n-i}...Sum_{k_n=0..1}
%F A000110 delta(k_1,k_2,...,k_i,...,k_n)
%F A000110 where delta(k_1,k_2,...,k_i,...,k_n) = 0 if any k_i > k_(i+1) and k_(i+1) <> 0
%F A000110 and delta(k_1,k_2,...,k_i,...,k_n) = 1 otherwise.
%F A000110 (End)
%F A000110 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). - _Milan Janjic_, Jul 08 2010
%F A000110 G.f. satisfies A(x) = (x/(1-x))*A(x/(1-x)) + 1. - _Vladimir Kruchinin_, Nov 28 2011
%F A000110 G.f.: 1 / (1 - x / (1 - 1*x / (1 - x / (1 - 2*x / (1 - x / (1 - 3*x / ... )))))). - _Michael Somos_, May 12 2012
%F A000110 a(n+1) = Sum_{m=0..n} Stirling2(n, m)*(m+1), n >= 0. Compare with the third formula for a(n) above. Here Stirling2 = A048993. - _Wolfdieter Lang_, Feb 03 2015
%F A000110 G.f.: (-1)^(1/x)*((-1/x)!/e + (!(-1-1/x))/x) where z! and !z are factorial and subfactorial generalized to complex arguments. - _Vladimir Reshetnikov_, Apr 24 2013
%F A000110 The following formulas were proposed during the period Dec 2011 - Oct 2013 by _Sergei N. Gladkovskii_: (Start)
%F A000110 E.g.f.: exp(exp(x)-1) = 1 + x/(G(0)-x); G(k) = (k+1)*Bell(k) + x*Bell(k+1) - x*(k+1)*Bell(k)*Bell(k+2)/G(k+1) (continued fraction).
%F A000110 G.f.: W(x) = (1-1/(G(0)+1))/exp(1); G(k) = x*k^2 + (3*x-1)*k - 2 + x - (k+1)*(x*k+x-1)^2/G(k+1); (continued fraction Euler's kind, 1-step).
%F A000110 G.f.: W(x) = (1 + G(0)/(x^2-3*x+2))/exp(1); G(k) = 1 - (x*k+x-1)/( ((k+1)!) - (((k+1)!)^2)*(1-x-k*x+(k+1)!)/( ((k+1)!)*(1-x-k*x+(k+1)!) - (x*k+2*x-1)*(1-2*x-k*x+(k+2)!)/G(k+1))); (continued fraction).
%F A000110 G.f.: A(x) = 1/(1 - x/(1-x/(1 + x/G(0)))); G(k) = x - 1 + x*k + x*(x-1+x*k)/G(k+1); (continued fraction, 1-step).
%F A000110 G.f.: -1/U(0) where U(k) = x*k - 1 + x - x^2*(k+1)/U(k+1); (continued fraction, 1-step).
%F A000110 G.f.: 1 + x/U(0) where U(k) = 1 - x*(k+2) - x^2*(k+1)/U(k+1); (continued fraction, 1-step).
%F A000110 G.f.: 1 + 1/(U(0) - x) where U(k) = 1 + x - x*(k+1)/(1 - x/U(k+1)); (continued fraction, 2-step).
%F A000110 G.f.: 1 + x/(U(0)-x) where U(k) = 1 - x*(k+1)/(1 - x/U(k+1)); (continued fraction, 2-step).
%F A000110 G.f.: 1/G(0) where G(k) = 1 - x/(1 - x*(2*k+1)/(1 - x/(1 - x*(2*k+2)/G(k+1) ))); (continued fraction).
%F A000110 G.f.: G(0)/(1+x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (continued fraction).
%F A000110 G.f.: -(1+2*x) * Sum_{k >= 0} x^(2*k)*(4*x*k^2-2*k-2*x-1) / ((2*k+1) * (2*x*k-1)) * A(k) / B(k) where A(k) = Product_{p=0..k} (2*p+1), B(k) = Product_{p=0..k} (2*p-1) * (2*x*p-x-1) * (2*x*p-2*x-1).
%F A000110 G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction).
%F A000110 G.f.: 1 + x*(S-1) where S = Sum_{k>=0} ( 1 + (1-x)/(1-x-x*k) )*(x/(1-x))^k/Product_{i=0..k-1} (1-x-x*i)/(1-x).
%F A000110 G.f.: (G(0) - 2)/(2*x-1) where G(k) = 2 - 1/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction).
%F A000110 G.f.: -G(0) where G(k) = 1 - (x*k - 2)/(x*k - 1 - x*(x*k - 1)/(x + (x*k - 2)/G(k+1) )); (continued fraction).
%F A000110 G.f.: G(0) where G(k) = 2 - (2*x*k - 1)/(x*k - 1 - x*(x*k - 1)/(x + (2*x*k - 1)/G(k+1) )); (continued fraction).
%F A000110 G.f.: (G(0) - 1)/(1+x) where G(k) = 1 + 1/(1-k*x)/(1-x/(x+1/G(k+1) )); (continued fraction).
%F A000110 G.f.: 1/(x*(1-x)*G(0)) - 1/x where G(k) = 1 - x/(x - 1/(1 + 1/(x*k-1)/G(k+1) )); (continued fraction).
%F A000110 G.f.: 1 + x/( Q(0) - x ) where Q(k) = 1 + x/( x*k - 1 )/Q(k+1); (continued fraction).
%F A000110 G.f.: 1+x/Q(0), where Q(k) = 1 - x - x/(1 - x*(k+1)/Q(k+1)); (continued fraction).
%F A000110 G.f.: 1/(1-x*Q(0)), where Q(k) = 1 + x/(1 - x + x*(k+1)/(x - 1/Q(k+1))); (continued fraction).
%F A000110 G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*(k+1))*(1-x*(k+2))/Q(k+1) ); (continued fraction).
%F A000110 (End)
%F A000110 a(n) ~ exp(exp(W(n))-n-1)*n^n/W(n)^(n+1/2), where W(x) is the Lambert W-function. - _Vladimir Reshetnikov_, Nov 01 2015
%F A000110 a(n) ~ n^n * exp(n/W(n)-1-n) / (sqrt(1+W(n)) * W(n)^n). - _Vaclav Kotesovec_, Nov 13 2015
%F A000110 From _Natalia L. Skirrow_, Apr 13 2025: (Start)
%F A000110 By taking logarithmic derivatives of the equivalent to Kotesovec's asymptotic for Bell polynomials at x=1, we obtain properties of the nth row of A008277 as a statistical distribution (where W=W(n),X=W(n)+1)
%F A000110 a(n+1)/a(n) ~ n/W + W/(2*(W+1)^2) is 1 more than the expectation.
%F A000110 (2*a(n+1)+a(n+2))/a(n) - (a(n+1)/a(n))^2 - a(n+2)/a(n+1) ~ n/(W*X)+1/(2*X^2)-3/(2*X^3)+1/X^4 is 1 more than the variance.
%F A000110 (This is a complete asymptotic characterisation, since they converge to normal distributions; see Harper, 1967)
%F A000110 (End)
%F A000110 a(n) are the coefficients in the asymptotic expansion of -exp(-1)*(-1)^x*x*Gamma(-x,0,-1), where Gamma(a,z0,z1) is the generalized incomplete Gamma function. - _Vladimir Reshetnikov_, Nov 12 2015
%F A000110 a(n) = 1 + floor(exp(-1) * Sum_{k=1..2*n} k^n/k!). - _Vladimir Reshetnikov_, Nov 13 2015
%F A000110 a(p^m) == m+1 (mod p) when p is prime and m >= 1 (see Lemma 3.1 in the Hurst/Schultz reference). - _Seiichi Manyama_, Jun 01 2016
%F A000110 a(n) = Sum_{k=0..n} hypergeom([1, -k], [], 1)*Stirling2(n+1, k+1) = Sum_{k=0..n} A182386(k)*Stirling2(n+1, k+1). - _Mélika Tebni_, Jul 02 2022
%F A000110 For n >= 1, a(n) = Sum_{i=0..n-1} a(i)*A074664(n-i). - _Davide Rotondo_, Apr 21 2024
%F A000110 a(n) is the n-th derivative of e^e^x divided by e at point x=0. - _Joan Ludevid_, Nov 05 2024
%e A000110 G.f. = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 + 4140*x^8 + ...
%e A000110 From Neven Juric, Oct 19 2009: (Start)
%e A000110 The a(4)=15 rhyme schemes for n=4 are
%e A000110 aaaa, aaab, aaba, aabb, aabc, abaa, abab, abac, abba, abbb, abbc, abca, abcb, abcc, abcd
%e A000110 The a(5)=52 rhyme schemes for n=5 are
%e A000110 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
%e A000110 (End)
%e A000110 From _Joerg Arndt_, Apr 30 2011: (Start)
%e A000110 Restricted growth strings (RGS):
%e A000110 For n=0 there is one empty string;
%e A000110 for n=1 there is one string [0];
%e A000110 for n=2 there are 2 strings [00], [01];
%e A000110 for n=3 there are a(3)=5 strings [000], [001], [010], [011], and [012];
%e A000110 for n=4 there are a(4)=15 strings
%e A000110 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].
%e A000110 These are one-to-one with the rhyme schemes (identify a=0, b=1, c=2, etc.).
%e A000110 (End)
%e A000110 Consider the set S = {1, 2, 3, 4}. The a(4) = 1 + 3 + 6 + 4 + 1 = 15 partitions are: P1 = {{1}, {2}, {3}, {4}}; P21 .. P23 = {{a,4}, S\{a,4}} with a = 1, 2, 3; P24 .. P29 = {{a}, {b}, S\{a,b}} with 1 <= a < b <= 4; P31 .. P34 = {S\{a}, {a}} with a = 1 .. 4; P4 = {S}. See the Bottomley link for a graphical illustration. - _M. F. Hasler_, Oct 26 2017
%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 A000110:= n-> add(Stirling2(n, k), k=0..n): seq(A000110(n), n=0..22); # version 3, from _Zerinvary Lajos_, Jun 28 2007
%p A000110 A000110 := n -> combinat[bell](n): # version 4, from _Peter Luschny_, Mar 30 2011
%p A000110 spec:= [S, {S=Set(U, card >= 1), U=Set(Z, card >= 1)}, labeled]: G:={P=Set(Set(Atom, card>0))}: combstruct[gfsolve](G, unlabeled, x): seq(combstruct[count]([P, G, labeled], size=i), i=0..22); # version 5, _Zerinvary Lajos_, Dec 16 2007
%p A000110 BellList := proc(m) local A, P, n; A := [1, 1]; P := [1]; for n from 1 to m - 2 do
%p A000110 P := ListTools:-PartialSums([A[-1], op(P)]); A := [op(A), P[-1]] od; A end: BellList(29); # _Peter Luschny_, Mar 24 2022
%t A000110 f[n_] := Sum[ StirlingS2[n, k], {k, 0, n}]; Table[ f[n], {n, 0, 40}] (* _Robert G. Wilson v_ *)
%t A000110 Table[BellB[n], {n, 0, 40}] (* _Harvey P. Dale_, Mar 01 2011 *)
%t A000110 B[0] = 1; B[n_] := 1/E Sum[k^(n - 1)/(k-1)!, {k, 1, Infinity}] (* _Dimitri Papadopoulos_, Mar 10 2015, edited by _M. F. Hasler_, Nov 30 2018 *)
%t A000110 BellB[Range[0,40]] (* _Eric W. Weisstein_, Aug 10 2017 *)
%t A000110 b[1] = 1; k = 1; Flatten[{1, Table[Do[j = k; k += b[m]; b[m] = j;, {m, 1, n-1}]; b[n] = k, {n, 1, 40}]}] (* _Vaclav Kotesovec_, Sep 07 2019 *)
%t A000110 Table[j! Coefficient[Series[Exp[Exp[x] - 1], {x, 0, 20}], x, j], {j, 0, 20}] (* _Nikolaos Pantelidis_, Feb 01 2023 *)
%t A000110 Table[(D[Exp[Exp[x]], {x, n}] /. x -> 0)/E, {n, 0, 20}] (* _Joan Ludevid_, Nov 05 2024 *)
%o A000110 (PARI) {a(n) = my(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))}; /* _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)=round(exp(-1)*suminf(k=0,1.0*k^n/k!)) \\ _Gottfried Helms_, Mar 30 2007 - WARNING! For illustration only: Gives silently a wrong result for n = 42 and an error for n > 42, with standard precision of 38 digits. - _M. F. Hasler_, Nov 30 2018
%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) Vec(serlaplace(exp(exp('x+O('x^66))-1))) \\ _Joerg Arndt_, May 26 2012
%o A000110 (PARI) A000110(n)=sum(k=0,n,stirling(n,k,2)) \\ _M. F. Hasler_, Nov 30 2018
%o A000110 (Sage) from sage.combinat.expnums import expnums2; expnums2(30, 1) # _Zerinvary Lajos_, Jun 26 2008
%o A000110 (Sage) [bell_number(n) for n in (0..40)] # _G. C. Greubel_, Jun 13 2019
%o A000110 (Python) # The objective of this implementation is efficiency.
%o A000110 # m -> [a(0), a(1), ..., a(m)] for m > 0.
%o A000110 def A000110_list(m):
%o A000110 A = [0 for i in range(m)]
%o A000110 A[0] = 1
%o A000110 R = [1, 1]
%o A000110 for n in range(1, m):
%o A000110 A[n] = A[0]
%o A000110 for k in range(n, 0, -1):
%o A000110 A[k-1] += A[k]
%o A000110 R.append(A[0])
%o A000110 return R
%o A000110 A000110_list(40) # _Peter Luschny_, Jan 18 2011
%o A000110 (Python)
%o A000110 # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
%o A000110 from itertools import accumulate
%o A000110 A000110, blist, b = [1,1], [1], 1
%o A000110 for _ in range(20):
%o A000110 blist = list(accumulate([b]+blist))
%o A000110 b = blist[-1]
%o A000110 A000110.append(b) # _Chai Wah Wu_, Sep 02 2014, updated _Chai Wah Wu_, Sep 19 2014
%o A000110 (Python)
%o A000110 from sympy import bell
%o A000110 print([bell(n) for n in range(27)]) # _Michael S. Branicky_, Dec 15 2021
%o A000110 (Python)
%o A000110 from functools import cache
%o A000110 @cache
%o A000110 def a(n, k=0): return int(n < 1) or k*a(n-1, k) + a(n-1, k+1)
%o A000110 print([a(n) for n in range(27)]) # _Peter Luschny_, Jun 14 2022
%o A000110 (Magma) [Bell(n): n in [0..40]]; // _Vincenzo Librandi_, Feb 07 2011
%o A000110 (Maxima) makelist(belln(n),n,0,40); /* _Emanuele Munarini_, Jul 04 2011 */
%o A000110 (Haskell)
%o A000110 type N = Integer
%o A000110 n_partitioned_k :: N -> N -> N
%o A000110 1 `n_partitioned_k` 1 = 1
%o A000110 1 `n_partitioned_k` _ = 0
%o A000110 n `n_partitioned_k` k = k * (pred n `n_partitioned_k` k) + (pred n `n_partitioned_k` pred k)
%o A000110 n_partitioned :: N -> N
%o A000110 n_partitioned 0 = 1
%o A000110 n_partitioned n = sum $ map (\k -> n `n_partitioned_k` k) $ [1 .. n]
%o A000110 -- _Felix Denis_, Oct 16 2012
%o A000110 (Haskell)
%o A000110 a000110 = sum . a048993_row -- _Reinhard Zumkeller_, Jun 30 2013
%o A000110 (Julia)
%o A000110 function a(n)
%o A000110 t = [zeros(BigInt, n+1) for _ in 1:n+1]
%o A000110 t[1][1] = 1
%o A000110 for i in 2:n+1
%o A000110 t[i][1] = t[i-1][i-1]
%o A000110 for j in 2:i
%o A000110 t[i][j] = t[i-1][j-1] + t[i][j-1]
%o A000110 end
%o A000110 end
%o A000110 return [t[i][1] for i in 1:n+1]
%o A000110 end
%o A000110 print(a(28)) # _Paul Muljadi_, May 07 2024
%o A000110 (Perl)
%o A000110 use bignum;sub a{my($n)=@_;my@t=map{[(0)x($n+1)]}0..$n;$t[0][0]=1;for my$i(1..$n){$t[$i][0]=$t[$i-1][$i-1];for my$j(1..$i){$t[$i][$j]=$t[$i-1][$j-1]+$t[$i][$j-1]}}return map{$t[$_][0]}0..$n-1}print join(", ",a(28)),"\n" # _Paul Muljadi_, May 08 2024
%Y A000110 Equals row sums of triangle A008277 (Stirling subset numbers).
%Y A000110 Partial sums give A005001. a(n) = A123158(n, 0).
%Y A000110 See A061462 for powers of 2 dividing a(n).
%Y A000110 Rightmost diagonal of triangle A121207. A144293 gives largest prime factor.
%Y A000110 Cf. A000045, A000108, A000166, A000204, A000255, A000311, A000296, A003422, A024716, A029761, A049020, A058692, A060719, A084423, A087650, A094262, A103293, A165194, A165196, A173110, A227840, A182386.
%Y A000110 Equals row sums of triangle A152432.
%Y A000110 Row sums, right and left borders of A212431.
%Y A000110 A diagonal of A011971. - _N. J. A. Sloane_, Jul 31 2012
%Y A000110 Diagonal of A102661. - _Manfred Boergens_, Mar 11 2025
%Y A000110 Cf. A054767 (period of this sequence mod n).
%Y A000110 Row sums are A048993. - _Wolfdieter Lang_, Oct 16 2014
%Y A000110 Sequences in the Erné (1974) paper: A000110, A000798, A001035, A001927, A001929, A006056, A006057, A006058, A006059.
%Y A000110 Bell polynomials B(n,x): A001861 (x=2), A027710 (x=3), A078944 (x=4), A144180 (x=5), A144223 (x=6), A144263 (x=7), A221159 (x=8).
%Y A000110 Cf. A243991 (sum of reciprocals), A085686 (inv. Euler Transf.).
%Y A000110 Cf. A000371, A003465, A186021.
%K A000110 core,nonn,easy,nice
%O A000110 0,3
%A A000110 _N. J. A. Sloane_
%E A000110 Edited by _M. F. Hasler_, Nov 30 2018