|
OFFSET
|
0,3
|
|
|
COMMENT
|
Number of partitions of an n-element set.
a(n-1) = number of nonisomorphic colorings of a map consisting of a row of n+1 adjacent regions. - David Wilson, Feb 22, 2005
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
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
Asymptotic expansion of (0!+1!+2!+...+n!)/n! = a(0)+a(1)/n+a(2)/n^2+... - Michael Somos, Aug 22 2004
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
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
|
|
REFERENCES
|
M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 205.
J. Balogh, B. Bollobas and D. Weinreich, A jump to the Bell numbers for hereditary graph properties, J. Combin. TheorySeries B, to appear (2006).
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.
E. T. Bell, The iterated exponential numbers, Ann. Math., 39 (1938), 539-557.
C. M. Bender, D. C. Brody and B. K. Meister, Quantum Field Theory of Partitions, J. Math. Phys., 40,7 (1999), 3239-45.
G. Birkhoff, Lattice Theory, Amer. Math. Soc., Revised Ed., 1961, p. 108, Ex. 1.
M. E. Cesaro, Sur une equation aux differences melees, Nouvelles Annales de Math. (3), Tome 4, (1885), 36--40.
C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 210.
N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, 1981, Sections 3.3. Case b and 6.1-6.3.
G. Dobinski, Summierung der Reihe Sum(n^m/n!) fuer m = 1, 2, 3, 4, 5, . . ., Grunert Archiv (Arch. f. Math. und Physik), 61 (1877) 333-336.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Flajolet, Philippe, and Schott, Rene, Nonoverlapping partitions, continued fractions, Bessel functions and a divergent series, European J. Combin. 11 (1990), no. 5, 421-432.
Martin Gardner, Fractal Music, Hypercards, and More (Freeman, 1992), Chapter 2.
H. W. Gould, Research bibliography of two special number sequences, Mathematica Monongaliae, Vol. 12, 1971.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., p. 493.
M. Klazar, Counting even and odd partitions, Amer. Math. Monthly, 110 (No. 6, 2003), 527-532.
M. Klazar, Bell numbers, their relatives, and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.
Christian Kramp, Der polynomische Lehrsatz (Leipzig: 1796), 113.
G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.
J. Levine and R. E. Dalton, Minimum periods, modulo p, of first-order Bell exponential integers, Math. Comp., 16 (1962), 416-423.
S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 14-15.
W. F. Lunnon, P. A. B. Pleasants and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979) 1-16.
N. S. Mendelsohn, Number of equivalence relations for n elements, Problem 4340, Amer. Math. Monthly 58 (1951), 46-48.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2, (2005), 215-224.
A. Murthy, Generalization of partition function, introducing Smarandache factor partition, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis , Phoenix; USA 2005. See Section 1.4,1.8.
M. Rayburn, On the Borel fields of a finite set, Proc. Amer. Math.. Soc., 19 (1968), 885-889.
C. Reid, The alternative life of E. T. Bell, Amer. Math. Monthly, 108 (No. 5, 2001), 393-402.
G.-C. Rota, The number of partitions of a set. Amer. Math. Monthly 71 1964 498-504.
R. P. Stanley, Enumerative Combinatorics, Cambridge; see Section 1.4 and Example 5.2.4.
|
|
LINKS
|
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
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).
R. M. Dickau, Bell number diagrams
John Fiorillo, GENJI-MON
H. Fripertinger, The Bell Numbers
Daniel L. Geisler, Combinatorics of Iterated Functions
A. Gertsch & A. M.Robert, Some congruences concerning the Bell numbers
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
Kazuhiro Kunii, Genji-koh no zu [Japanese page illustrating a(5) = 52]
A. F. Labossiere, Sobalian Coefficients.
A. F. Labossiere, Miscellaneous.
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.
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 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.
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.
E. W. Weisstein, Link to a section of The World of Mathematics (1).
E. W. Weisstein, Link to a section of The World of Mathematics (2).
E. W. Weisstein, Link to a section of The World of Mathematics (3).
Eric Weisstein's World of Mathematics, Bell Triangle
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, pp. 21ff.
Index entries for sequences related to rooted trees
Index entries for "core" sequences
|
|
FORMULA
|
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 (sobal(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 (abmt(AT)wanadoo.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 (abmt(AT)wanadoo.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
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 (sobal(AT)laposte.net), Feb 22 2005
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
|
|
MAPLE
|
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
with(combstruct); spec := [S, {S=Set(U, card >= 1), U=Set(Z, card >= 1)}, labeled]; [seq(combstruct[count](spec, size=n), n=0..40)];
|
|
MATHEMATICA
|
f[n_] := Sum[ StirlingS2[n, k], {k, 1, n}]; Table[ f[n], {n, 0, 21}] (from Robert G. Wilson v)
|
|
PROGRAM
|
(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)) (from Michael Somos)
(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 */
|
|
CROSSREFS
|
Partial sums give A005001. Cf. A049020. See A061462 for powers of 2 dividing a(n).
Cf. A000311, A103293.
Cf. A005001, A087650, A029761, A024716, A000296, A058692, A060719.
Cf. A008277, A000166, A000255, A000108, A000045, A000204.
Cf. A094262, A008277, A005001, A003422, A000166, A000204, A000045, A000108.
Sequence in context: A099262 A108305 A099263 this_sequence A107589 A006790 A007548
Adjacent sequences: A000107 A000108 A000109 this_sequence A000111 A000112 A000113
|
|
KEYWORD
|
core,nonn,easy,nice
|
|
AUTHOR
|
njas
|
|
