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%I M2950 N1190 #467 Mar 30 2024 09:38:08
%S 1,1,3,13,73,501,4051,37633,394353,4596553,58941091,824073141,
%T 12470162233,202976401213,3535017524403,65573803186921,
%U 1290434218669921,26846616451246353,588633468315403843,13564373693588558173,327697927886085654441,8281153039765859726341
%N Number of "sets of lists": number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset.
%C Determinant of n X n matrix M=[m(i,j)] where m(i,i)=i, m(i,j)=1 if i > j, m(i,j)=i-j if j > i. - _Vladeta Jovovic_, Jan 19 2003
%C With p(n) = the number of integer partitions of n, d(i) = the number of different parts 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 Product_{j=1..d(i)} = product over j, one has: a(n) = Sum_{i=1..p(n)} n!/(Product_{j=1..d(i)} m(i,j)!). - _Thomas Wieder_, May 18 2005
%C Consider all n! permutations of the integer sequence [n] = 1,2,3,...,n. The i-th permutation, i=1,2,...,n!, consists of Z(i) permutation cycles. Such a cycle has the length lc(i,j), j=1,...,Z(i). For a given permutation we form the product of all its cycle lengths Product_{j=1..Z(i)} lc(i,j). Furthermore, we sum up all such products for all permutations of [n] which gives Sum_{i=1..n!} Product_{j=1..Z(i)} lc(i,j) = A000262(n). For n=4 we have Sum_{i=1..n!} Product_{j=1..Z(i)} lc(i,j) = 1*1*1*1 + 2*1*1 + 3*1 + 2*1*1 + 3*1 + 2*1 + 3*1 + 4 + 3*1 + 4 + 2*2 + 2*1*1 + 3*1 + 4 + 3*1 + 2*1*1 + 2*2 + 4 + 2*2 + 4 + 3*1 + 2*1*1 + 3*1 + 4 = 73 = A000262(4). - _Thomas Wieder_, Oct 06 2006
%C For a finite set S of size n, a chain gang G of S is a partially ordered set (S,<=) consisting only of chains. The number of chain gangs of S is a(n). For example, with S={a, b}and n=2, there are a(2)=3 chain gangs of S, namely, {(a,a),(b,b)}, {(a,a),(a,b),(b,b)} and {(a,a),(b,a),(b,b)}. - _Dennis P. Walsh_, Feb 22 2007
%C (-1)*A000262 with the first term set to 1 and A084358 form a reciprocal pair under the list partition transform and associated operations described in A133314. Cf. A133289. - _Tom Copeland_, Oct 21 2007
%C Consider the distribution of n unlabeled elements "1" onto n levels where empty levels are allowed. In addition, the empty levels are labeled. Their names are 0_1, 0_2, 0_3, etc. This sequence gives the total number of such distributions. If the empty levels are unlabeled ("0"), then the answer is A001700. Let the colon ":" separate two levels. Then, for example, for n=3 we have a(3)=13 arrangements: 111:0_1:0_2, 0_1:111:0_2, 0_1:0_2:111, 111:0_2:0_1, 0_2:111:0_1, 0_2:0_1:111, 11:1:0, 11:0:1, 0:11:1, 1:11:0, 1:0:11, 0:1:11, 1:1:1. - _Thomas Wieder_, May 25 2008
%C Row sums of exponential Riordan array [1,x/(1-x)]. - _Paul Barry_, Jul 24 2008
%C a(n) is the number of partitions of [n] (A000110) into lists of noncrossing sets. For example, a(3)=3 counts 12, 1-2, 2-1 and a(4) = 73 counts the 75 partitions of [n] into lists of sets (A000670) except for 13-24, 24-13 which fail to be noncrossing. - _David Callan_, Jul 25 2008
%C a(i-j)/(i-j)! gives the value of the non-null element (i, j) of the lower triangular matrix exp(S)/exp(1), where S is the lower triangular matrix - of whatever dimension - having all its (non-null) elements equal to one. - _Giuliano Cabrele_, Aug 11 2008, Sep 07 2008
%C a(n) is also the number of nilpotent partial one-one bijections (of an n-element set). Equivalently, it is the number of nilpotents in the symmetric inverse semigroup (monoid). - _Abdullahi Umar_, Sep 14 2008
%C A000262 is the exp transform of the factorial numbers A000142. - _Thomas Wieder_, Sep 10 2008
%C If n is a positive integer then the infinite continued fraction (1+n)/(1+(2+n)/(2+(3+n)/(3+...))) converges to the rational number A052852(n)/A000262(n). - David Angell (angell(AT)maths.unsw.edu.au), Dec 18 2008
%C _Vladeta Jovovic_'s formula dated Sep 20 2006 can be restated as follows: a(n) is the n-th ascending factorial moment of the Poisson distribution with parameter (mean) 1. - Shai Covo (green355(AT)netvision.net.il), Jan 25 2010
%C The umbral exponential generating function for a(n) is (1-x)^{-B}. In other words, write (1-x)^{-B} as a power series in x whose coefficients are polynomials in B, and then replace B^k with the Bell number B_k. We obtain a(0) + a(1)x + a(2)x^2/2! + ... . - _Richard Stanley_, Jun 07 2010
%C a(n) is the number of Dyck n-paths (A000108) with its peaks labeled 1,2,...,k in some order, where k is the number of peaks. For example a(2)=3 counts U(1)DU(2)D, U(2)DU(1)D, UU(1)DD where the label at each peak is in parentheses. This is easy to prove using generating functions. - _David Callan_, Aug 23 2011
%C a(n) = number of permutations of the multiset {1,1,2,2,...,n,n} such that for 1 <= i <= n, all entries between the two i's exceed i and if any such entries are present, they include n. There are (2n-1)!! permutations satisfying the first condition, and for example: a(3)=13 counts all 5!!=15 of them except 331221 and 122133 which fail the second condition. - _David Callan_, Aug 27 2014_
%C a(n) is the number of acyclic, injective functions from subsets of [n] to [n]. Let subset D of [n] have size k. The number of acyclic, injective functions from D to [n] is (n-1)!/(n-k-1)! and hence a(n) = Sum_{k=0..n-1} binomial(n,k)*(n-1)!/(n-k-1)!. - _Dennis P. Walsh_, Nov 05 2015
%C a(n) is the number of acyclic, injective, labeled directed graphs on n vertices with each vertex having outdegree at most one. - _Dennis P. Walsh_, Nov 05 2015
%C For n > 0, a(n) is the number of labeled-rooted skinny-tree forests on n nodes. A skinny tree is a tree in which each vertex has at most one child. Let k denote the number of trees. There are binomial(n,k) ways to choose the roots, binomial(n-1,k-1) ways to choose the number of descendants for each root, and (n-k)! ways to permute those descendants. Summing over k, we obtain a(n) = Sum_{k=1..n} C(n,k)*C(n-1,k-1)*(n-k)!. - _Dennis P. Walsh_, Nov 10 2015
%C This is the Sheffer transform of the Bell numbers A000110 with the Sheffer matrix S = |Stirling1| = (1, -log(1-x)) = A132393. See the e.g.f. formula, a Feb 21 2017 comment on A048993, and R. Stanley's Jun 07 2010 comment above. - _Wolfdieter Lang_, Feb 21 2017
%C All terms = {1, 3} mod 10. - _Muniru A Asiru_, Oct 01 2017
%C We conjecture that for k = 2,3,4,..., the difference a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k is periodic with period dividing k. - _Peter Bala_, Nov 14 2017
%C The above conjecture is true - see the Bala link. - _Peter Bala_, Jan 20 2018
%C The terms of this sequence can be derived from the numerators of the fractions, produced by the recursion: b(0) = 1, b(n) = 1 + ((n-1) * b(n-1)) / (n-1 + b(n-1)) for n > 0. The denominators give A002720. - _Dimitris Valianatos_, Aug 01 2018
%C a(n) is the number of rooted labeled forests on n nodes that avoid the patterns 213, 312, and 123. It is also the number of rooted labeled forests that avoid 312, 213, and 132, as well as the number of rooted labeled forests that avoid 132, 213, and 321. - _Kassie Archer_, Aug 30 2018
%C For n > 0, a(n) is the number of partitions of [2n-1] whose nontrivial blocks are of type {a,b}, with a <= n and b > n. In fact, for n > 0, a(n) = A056953(2n-1). - _Francesca Aicardi_, Nov 03 2022
%C For n > 0, a(n) is the number of ways to split n people into nonempty groups, have each group sit around a circular table, and select one person from each table (where two seating arrangements are considered identical if each person has the same left neighbors in both of them). - _Enrique Navarrete_, Oct 01 2023
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 194.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000262/b000262.txt">Table of n, a(n) for n = 0..444</a> (first 101 terms from T. D. Noe)
%H A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, <a href="http://arxiv.org/abs/1402.2960">Bell polynomials in combinatorial Hopf algebras</a>, arXiv preprint arXiv:1402.2960 [math.CO], 2014.
%H Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, <a href="https://arxiv.org/abs/2210.17461">Ramified inverse and planar monoids</a>, arXiv:2210.17461 [math.RT], 2022.
%H David Angell, <a href="http://dx.doi.org/10.1016/j.jnt.2009.12.003">A family of continued fractions</a>, Journal of Number Theory, Volume 130, Issue 4, April 2010, Pages 904-911. Section 2.
%H Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry3/barry100r.html">The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms</a>, J. Int. Seq. 13 (2010) # 10.8.4, example 4.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry4/barry122.html">Exponential Riordan Arrays and Permutation Enumeration</a>, J. Int. Seq. 13 (2010) # 10.9.1, example 6.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry1/barry97r2.html">Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms</a>, J. Int. Seq. 14 (2011) # 11.2.2, example 18.
%H Paul Barry, <a href="http://arxiv.org/abs/1105.3044">Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays</a>, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry5/barry112.html">J. Int. Seq. 14 (2011) 11.6.7</a>.
%H Andreas Bärtschi, Daniel Graf, and Paolo Penna, <a href="http://drops.dagstuhl.de/opus/volltexte/2017/7889/">Truthful Mechanisms for Delivery with Agents</a>, 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems, OASIcs, Volume 59, 2017.
%H Beáta Bényi, Miguel Méndez, and José L. Ramirez, <a href="https://arxiv.org/abs/2006.02794">Generalized Ordered Set Partitions</a>, arXiv:2006.02794 [math.CO], 2020.
%H P. Blasiak, A. Horzela, K. A. Penson, G. H. E. Duchamp and A. I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0501155">Boson normal ordering via substitutions and Sheffer-type polynomials</a>, arXiv:quant-ph/0501155, 2005.
%H 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>, arXiv:quant-ph/0212072, 2002.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, <a href="https://arxiv.org/abs/1812.00316">A Conjectured Integer Sequence Arising From the Exponential Integral</a>, arXiv:1812.00316 [math.NT], 2018.
%H J.-P. Bultel, A, Chouria, J.-G. Luque and O. Mallet, <a href="http://hal.archives-ouvertes.fr/docs/00/79/37/88/PDF/Polya_arxiv.pdf">Word symmetric functions and the Redfield-Polya theorem</a>, HAL Id: hal-00793788, 2013.
%H David Callan, <a href="http://arxiv.org/abs/0711.4841">Sets, Lists and Noncrossing Partitions</a>, arXiv:0711.4841 [math.CO], 2007-2008.
%H David Callan and Emeric Deutsch, <a href="http://arxiv.org/abs/1112.3639">The Run Transform</a>, Discrete Math. 312 (2012), no. 19, 2927-2937, arXiv:1112.3639 [math.CO], 2011.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Clintin P. Davis-Stober, Jean-Paul Doignon, Samuel Fiorini, François Glineur, and Michel Regenwetter, <a href="https://arxiv.org/abs/1710.02679">Extended Formulations for Order Polytopes through Network Flows</a>, arXiv:1710.02679 [math.OC], 2017.
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 125.
%H S. Garrabrant and I. Pak, <a href="http://www.math.ucla.edu/~pak/papers/Dfinite12.pdf">Words in linear groups, computability and p-recursiveness</a>, 2015.
%H Stefan Gerhold, <a href="http://www.emis.de/journals/INTEGERS/papers/l44/l44.Abstract.html">Counting finite languages by total word length</a>, INTEGERS 11 (2011), #A44.
%H Bai-Ni Guo and Feng Qi, <a href="http://dx.doi.org/10.14419/gjma.v2i4.3310">An Explicit Formula for Bell Numbers in Terms of Stirling Numbers and Hypergeometric Functions</a>, Global Journal of Mathematical Analysis, 2 (No. 4, 2014), DOI: 10.14419/gjma.v2i4.3310 (Warning, this Journal is run by the 'Science Publishing Corporation', which is listed in Jeffrey Beall's <a href="http://scholarlyoa.com/publishers/">List of predatory publishers</a>).
%H Bai-Ni Guo and Feng Qi, <a href="http://arxiv.org/abs/1402.2361">An Explicit Formula for Bell Numbers in Terms of Stirling Numbers and Hypergeometric Functions</a>, arXiv:1402.2361 [math.CO], 2014.
%H T.-X. He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/He/he51.html">A symbolic operator approach to power series transformation-expansion formulas</a>, JIS 11 (2008) 08.2.7.
%H A. Hennessy and P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry6/barry161.html">Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials</a>, J. Int. Seq. 14 (2011) # 11.8.2.
%H F. Hivert, J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0605262">Commutative combinatorial Hopf algebras</a>, arXiv:math/0605262 [math.CO], 2006.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=40">Encyclopedia of Combinatorial Structures 40</a>
%H Salvador Jacobi, <a href="http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=6799">Planning in Multi-Agent Systems</a>, Thesis, Technical University of Denmark, Department of Applied Mathematics and Computer Science, 2800 Kongens Lyngby, Denmark, 2014.
%H D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, arXiv:math/9207221 [math.CA], 1992; The Mathematica J., 2 (1992), 67-78.
%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1081/AGB-120038637">On the number of nilpotents in the partial symmetric semigroup</a>, Comm. Algebra 32 (2004), 3017-3023.
%H Victor Meally, <a href="/A002868/a002868.pdf">Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.</a>
%H T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
%H Norihiro Nakashima and Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/1510.03033">On composition polynomials</a>, arXiv:1510.03033 [math.CO], 11 October 2015.
%H OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%H Satya R. T. Peddada, Daniel R. Herber, Herschel C. Pangborn, Andrew G. Alleyne, and James T. Allison, <a href="https://doi.org/10.1115/1.4043203">Optimal Flow Control and Single Split Architecture Exploration for Fluid-Based Thermal Management</a>, J. Mech. Des. (2019) 141(8), Paper No. MD-18-1843, 083401.
%H 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>, J. Phys. A 37 (2004), 3475-3487.
%H Robert A. Proctor, <a href="http://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way For Counting Partitions!</a>, arXiv:math.CO/0606404, Jan 05 2007.
%H Feng Qi, <a href="https://www.researchgate.net/profile/Feng_Qi/publication/279750659">On sum of the Lah numbers and zeros of the Kummer confluent hypergeometric function</a>, Research Gate, 2015.
%H Feng Qi, <a href="http://dx.doi.org/10.1007/s00009-015-0655-7">An Explicit Formula for the Bell Numbers in Terms of the Lah and Stirling Numbers</a>, Mediterranean Journal of Mathematics, November 2015, DOI: 10.1007/s00009-015-0655-7.
%H John Riordan, <a href="/A000262/a000262_1.pdf">Letter to N. J. A. Sloane, Oct. 1970</a>
%H John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.
%H Mark A. Shattuck and Carl G. Wagner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Shattuck2/shattuck44.html">Parity Theorems for Statistics on Lattice Paths and Laguerre Configurations</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.1.
%H M. Shattuck, <a href="http://arxiv.org/abs/1401.6588">Combinatorial proofs of some Bell number formulas</a>, arXiv preprint arXiv:1401.6588 [math.CO], 2014.
%H M. Shattuck, <a href="http://arxiv.org/abs/1412.8721">Generalized r-Lah numbers</a>, arXiv preprint arXiv:1412.8721 [math.CO], 2014.
%H Tomi Silander, Janne Leppä-aho, Elias Jääsaari, and Teemu Roos, <a href="https://www.cs.helsinki.fi/u/ttonteri/pub/aistats2018.pdf">Quotient Normalized Maximum Likelihood Criterion for Learning Bayesian Network Structures</a>, International Conference on Artificial Intelligence and Statistics, 2018.
%H N. J. A. Sloane and Thomas Wieder, <a href="https://arxiv.org/abs/math/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
%H A. Umar, <a href="http://www.mathnet.ru/adm33">Some combinatorial problems in the theory of symmetric ...</a>, Algebra Disc. Math. 9 (2010) 115-126
%H Thomas Wieder, <a href="/A000262/a000262.txt">Further comments on this sequence</a>
%H Y. Yakubovich, <a href="http://dx.doi.org/10.1016/j.jcta.2012.03.002">Ergodicity of multiplicative statistics</a>, Journal of Combinatorial Theory, Series A 119 (2012) 1250-1279, <a href="http://arxiv.org/abs/0901.4655">alternative copy</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F D-finite with recurrence: a(n) = (2*n-1)*a(n-1) - (n-1)*(n-2)*a(n-2).
%F E.g.f.: exp( x/(1-x) ).
%F a(n) = Sum_{k=0..n} |A008275(n,k)| * A000110(k). - _Vladeta Jovovic_, Feb 01 2003
%F a(n) = (n-1)!*LaguerreL(n-1,1,-1) for n >= 1. - _Vladeta Jovovic_, May 10 2003
%F Representation as n-th moment of a positive function on positive half-axis, in Maple notation: a(n) = integral(x^n*exp(-x-1)*BesselI(1, 2*x^(1/2))/x^(1/2), x =0..infinity), n=1, 2... - _Karol A. Penson_, Dec 04 2003
%F a(n) = Sum_{k=0..n} A001263(n, k)*k!. - _Philippe Deléham_, Dec 10 2003
%F a(n) = n! Sum_{j=0..n-1} binomial(n-1, j)/(j+1)!, for n > 0. - Herbert S. Wilf, Jun 14 2005
%F Asymptotic expansion for large n: a(n) -> (0.4289*n^(-1/4) + 0.3574*n^(-3/4) - 0.2531*n^(-5/4) + O(n^(-7/4)))*(n^n)*exp(-n + 2*sqrt(n)). - _Karol A. Penson_, Aug 28 2002
%F Minor part of this asymptotic expansion is wrong! Right is (in closed form): a(n) ~ n^(n-1/4)*exp(-1/2+2*sqrt(n)-n)/sqrt(2) * (1 - 5/(48*sqrt(n)) - 95/(4608*n)), numerically a(n) ~ (0.42888194248*n^(-1/4) - 0.0446752023417*n^(-3/4) - 0.00884196713*n^(-5/4) + O(n^(-7/4))) *(n^n)*exp(-n+2*sqrt(n)). - _Vaclav Kotesovec_, Jun 02 2013
%F a(n) = exp(-1)*Sum_{m>=0} [m]^n/m!, where [m]^n = m*(m+1)*...*(m+n-1) is the rising factorial. - _Vladeta Jovovic_, Sep 20 2006
%F Recurrence: D(n,k) = D(n-1,k-1) + (n-1+k) * D(n-1,k) n >= k >= 0; D(n,0)=0. From this, D(n,1) = n! and D(n,n)=1; a(n) = Sum_{i=0..n} D(n,i). - Stephen Dalton (StephenMDalton(AT)gmail.com), Jan 05 2007
%F Proof: Notice two distinct subsets of the lists for [n]: 1) n is in its own list, then there are D(n-1,k-1); 2) n is in a list with other numbers. Denoting the separation of lists by |, it is not hard to see n has (n-1+k) possible positions, so (n-1+k) * D(n-1,k). - Stephen Dalton (StephenMDalton(AT)gmail.com), Jan 05 2007
%F Define f_1(x), f_2(x), ... such that f_1(x) = exp(x), f_{n+1}(x) = (d/dx)(x^2*f_n(x)), for n >= 2. Then a(n-1) = exp(-1)*f_n(1). - _Milan Janjic_, May 30 2008
%F a(n) = (n-1)! * Sum_{k=1..n} (a(n-k)*k!)/((n-k)!*(k-1)!), where a(0)=1. - _Thomas Wieder_, Sep 10 2008
%F a(n) = exp(-1)*n!*M(n+1,2,1), n >= 1, where M (=1F1) is the confluent hypergeometric function of the first kind. - Shai Covo (green355(AT)netvision.net.il), Jan 20 2010
%F a(n) = n!* A067764(n) / A067653(n). - _Gary Detlefs_, May 15 2010
%F a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A000110, A049118, A049119 and A049120. - _Peter Bala_, Nov 25 2011
%F From _Sergei N. Gladkovskii_, Nov 17 2011, Aug 02 2012, Dec 11 2012, Jan 27 2013, Jul 31 2013, Dec 25 2013: (Start)
%F Continued fractions:
%F E.g.f.: Q(0) where Q(k) = 1+x/((1-x)*(2k+1)-x*(1-x)*(2k+1)/(x+(1-x)*(2k+2)/Q(k+1))).
%F E.g.f.: 1 + x/(G(0)-x) where G(k) = (1-x)*k + 1 - x*(1-x)*(k+1)/G(k+1); (Euler's 1st kind, 1-step).
%F E.g.f.: exp(x/(1-x)) = 4/(2-(x/(1-x))*G(0))-1 where G(k) = 1 - x^2/(x^2 + 4*(1-x)^2*(2*k+1)*(2*k+3)/G(k+1) ).
%F E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + 1/(k+1)/(1-x)/(1-x/(x+1/E(k+1) )).
%F E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - x/(x + (k+1)*(1-x)/E(k+1) )).
%F E.g.f.: E(0)-1, where E(k) = 2 + x/( (2*k+1)*(1-x) - x/E(k+1) ).
%F (End)
%F E.g.f.: Product {n >= 1} ( (1 + x^n)/(1 - x^n) )^( phi(2*n)/(2*n) ), where phi(n) = A000010(n) is the Euler totient function. Cf. A088009. - _Peter Bala_, Jan 01 2014
%F a(n) = n!*hypergeom([1-n],[2],-1) for n >= 1. - _Peter Luschny_, Jun 05 2014
%F a(n) = (-1)^(n-1)*KummerU(1-n,2,-1). - _Peter Luschny_, Sep 17 2014
%F a(n) = hypergeom([-n+1, -n], [], 1) for n >= 0. - _Peter Luschny_, Apr 08 2015
%F E.g.f.: Product_{k>0} exp(x^k). - _Franklin T. Adams-Watters_, May 11 2016
%F 0 = a(n)*(18*a(n+2) - 93*a(n+3) + 77*a(n+4) - 17*a(n+5) + a(n+6)) + a(n+1)*(9*a(n+2) - 80*a(n+3) + 51*a(n+4) - 6*a(n+5)) + a(n+2)*(3*a(n+2) - 34*a(n+3) + 15*a(n+4)) + a(n+3)*(-10*a(n+3)) if n >= 0. - _Michael Somos_, Feb 27 2017
%F G.f. G(x)=y satisfies a differential equation: (1-x)*y-2*(1-x)*x^2*y'+x^4*y''=1. - _Bradley Klee_, Aug 13 2018
%F a(n) = n! * LaguerreL(n, -1, -1) = c_{n}(n-1; -1) where c_{n}(x; a) are the Poisson - Charlier polynomials. - _G. C. Greubel_, Feb 23 2021
%F 3 divides a(3*n-1); 9 divides a(9*n-1); 11 divides a(11*n-1). - _Peter Bala_, Mar 26 2022
%F For n > 0, a(n) = Sum_{k=0..n-1}*k!*C(n-1,k)*C(n,k). - _Francesca Aicardi_, Nov 03 2022
%F For n > 0, a(n) = (n-1)! * (Sum_{i=0..n-1} A002720(i) / i!). - _Werner Schulte_, Mar 29 2024
%e Illustration of first terms as sets of ordered lists of the first n integers:
%e a(1) = 1 : (1)
%e a(2) = 3 : (12), (21), (1)(2).
%e a(3) = 13 : (123) (6 ways), (12)(3) (2*3 ways) (1)(2)(3) (1 way)
%e a(4) = 73 : (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (2*2*3 ways), (12)(3)(4) (2*6 ways), (1)(2)(3)(4) (1 way).
%e :
%e G.f. = 1 + x + 3*x^2 + 13*x^3 + 73*x^4 + 501*x^4 + 4051*x^5 + 37633*x^6 + 394353*x^7 + ...
%p A000262 := proc(n) option remember: if n=0 then RETURN(1) fi: if n=1 then RETURN(1) fi: (2*n-1)*procname(n-1) - (n-1)*(n-2)*procname(n-2) end proc:
%p for n from 0 to 20 do printf(`%d,`,a(n)) od:
%p spec := [S, {S = Set(Prod(Z,Sequence(Z)))}, labeled]; [seq(combstruct[count](spec, size=n), n=0..40)];
%p with(combinat):seq(sum(abs(stirling1(n, k))*bell(k), k=0..n), n=0..18); # _Zerinvary Lajos_, Nov 26 2006
%p B:=[S,{S = Set(Sequence(Z,1 <= card),card <=13)},labelled]: seq(combstruct[count](B, size=n), n=0..19);# _Zerinvary Lajos_, Mar 21 2009
%p a := n -> `if`(n=0, 1, n!*hypergeom([1 - n], [2], -1)): seq(simplify(a(n)), n=0..19); # _Peter Luschny_, Jun 05 2014
%t Range[0, 19]! CoefficientList[ Series[E^(x/(1-x)), {x, 0, 19}], x] (* _Robert G. Wilson v_, Apr 04 2005 *)
%t a[ n_]:= If[ n<0, 0, n! SeriesCoefficient[ Exp[x/(1-x)], {x, 0, n}]]; (* _Michael Somos_, Jul 19 2005 *)
%t a[n_]:=If[n==0, 1, n! Sum[Binomial[n-1, j]/(j+1)!, {j, 0, n-1}]]; Table[a[n], {n, 0, 30}] (* Wilf *)
%t a[0] = 1; a[n_]:= n!*Hypergeometric1F1[n+1, 2, 1]/E; Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Jun 18 2012, after Shai Covo *)
%t Table[Sum[BellY[n, k, Range[n]!], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%t a[ n_]:= If[ n<0, 0, n! SeriesCoefficient[ Product[ QPochhammer[x^k]^(-MoebiusMu[k]/k), {k, n}], {x, 0, n}]]; (* _Michael Somos_, Jun 02 2019 *)
%t Table[n!*LaguerreL[n, -1, -1], {n, 0, 30}] (* _G. C. Greubel_, Feb 23 2021 *)
%t RecurrenceTable[{a[n] == (2*n - 1)*a[n-1] - (n-1)*(n-2)*a[n-2], a[0] == 1, a[1] == 1}, a, {n, 0, 30}] (* _Vaclav Kotesovec_, Jul 21 2022 *)
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( x / (1 - x) + x * O(x^n)), n))}; /* _Michael Somos_, Feb 10 2005 */
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( prod( k=1, n, eta(x^k + x * O(x^n))^( -moebius(k) / k)), n))}; /* _Michael Somos_, Feb 10 2005 */
%o (PARI) {a(n) = s = 1; for(k = 1, n-1, s = 1 + k * s / (k + s)); return( numerator(s))}; /* _Dimitris Valianatos_, Aug 03 2018 */
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^(-eulerphi(k) / k)), n))}; /* _Michael Somos_, Jun 02 2019 */
%o (PARI) a(n) = (n-1)!*pollaguerre(n-1,1,-1); \\ _Michel Marcus_, Feb 23 2021
%o (Maxima) makelist(sum(abs(stirling1(n,k))*belln(k),k,0,n),n,0,24); /* _Emanuele Munarini_, Jul 04 2011 */
%o (Maxima) makelist(hypergeometric([-n+1,-n],[],1),n,0,12); /* _Emanuele Munarini_, Sep 27 2016 */
%o (Haskell)
%o a000262 n = a000262_list !! n
%o a000262_list = 1 : 1 : zipWith (-)
%o (tail $ zipWith (*) a005408_list a000262_list)
%o (zipWith (*) a002378_list a000262_list)
%o -- _Reinhard Zumkeller_, Mar 06 2014
%o (Sage)
%o A000262 = lambda n: hypergeometric([-n+1, -n], [], 1)
%o [simplify(A000262(n)) for n in (0..19)] # _Peter Luschny_, Apr 08 2015
%o (GAP)
%o a:=[1,1];; for n in [3..10^2] do a[n]:=(2*n-3)*a[n-1]-(n-2)*(n-3)*a[n-2]; od; A000262:=a; # _Muniru A Asiru_, Oct 01 2017
%o (Magma) I:=[1,3]; [1] cat [n le 2 select I[n] else (2*n-1)*Self(n-1) - (n-1)*(n-2)*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 14 2019
%o (Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), -1): n in [0..30]]; // _G. C. Greubel_, Feb 23 2021
%o (Python)
%o from sympy import hyper, hyperexpand
%o def A000262(n): return hyperexpand(hyper((-n+1, -n), [], 1)) # _Chai Wah Wu_, Jan 14 2022
%Y Row sums of A271703 and for n >= 1 of A008297. Unsigned row sums of A111596.
%Y A002868 is maximal element of the n-th row of A271703 and for n >= 1 of A008297.
%Y Main diagonal of A257740 and of A319501.
%Y Cf. A001263, A001700, A002378, A005408, A066668, A056953, A002720.
%Y Cf. A000110, A052852, A082579, A132393, A255807, A255819, A318976.
%K nonn,easy,core,nice
%O 0,3
%A _N. J. A. Sloane_
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