

A000262


Number of "sets of lists": number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset.
(Formerly M2950 N1190)


191



1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553, 58941091, 824073141, 12470162233, 202976401213, 3535017524403, 65573803186921, 1290434218669921, 26846616451246353, 588633468315403843, 13564373693588558173, 327697927886085654441, 8281153039765859726341
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

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)=ij if j > i.  Vladeta Jovovic, Jan 19 2003
a(n) = Sum_{k=0..n} A008275(n,k) * A000110(k).  Vladeta Jovovic, Feb 01 2003
a(n) = (n1)!*LaguerreL(n1,1,1) for n >= 1.  Vladeta Jovovic, May 10 2003
With p(n) = the number of integer partitions of n, d(i) = the number of different parts of the ith partition of n, m(i,j) = multiplicity of the jth part of the ith 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
Consider all n! permutations of the integer sequence [n] = 1,2,3,...,n. The ith 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
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
(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
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
Row sums of exponential Riordan array [1,x/(1x)].  Paul Barry, Jul 24 2008
a(n) is the number of partitions of [n] (A000110) into lists of noncrossing sets. For example, a(3)=3 counts 12, 12, 21 and a(4) = 73 counts the 75 partitions of [n] into lists of sets (A000670) except for 1324, 2413 which fail to be noncrossing.  David Callan, Jul 25 2008
a(ij)/(ij)! gives the value of the nonnull 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 (nonnull) elements equal to one.  Giuliano Cabrele, Aug 11 2008, Sep 07 2008
a(n) is also the number of nilpotent partial oneone bijections (of an nelement set). Equivalently, it is the number of nilpotents in the symmetric inverse semigroup (monoid).  Abdullahi Umar, Sep 14 2008
A000262 is the exp transform of the factorial numbers A000142.  Thomas Wieder, Sep 10 2008
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
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
Vladeta Jovovic's formula dated Sep 20 2006 can be restated as follows: a(n) is the nth ascending factorial moment of the Poisson distribution with parameter (mean) 1.  Shai Covo (green355(AT)netvision.net.il), Jan 25 2010
a(n) = n!* A067764(n) / A067653(n).  Gary Detlefs, May 15 2010
The umbral exponential generating function for a(n) is (1x)^{B}. In other words, write (1x)^{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
a(n) is the number of Dyck npaths (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
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 (2n1)!! 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_
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 (n1)!/(nk1)! and hence a(n) = Sum_{k=0..n1} binomial(n,k)*(n1)!/(nk1)!.  Dennis P. Walsh, Nov 05 2015
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
For n > 0, a(n) is the number of labeledrooted skinnytree 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(n1,k1) ways to choose the number of descendants for each root, and (nk)! ways to permute those descendants. Summing over k, we obtain a(n) = Sum_{k=1..n} C(n,k)*C(n1,k1)*(nk)!.  Dennis P. Walsh, Nov 10 2015
This is the Sheffer transform of the Bell numbers A000110 with the Sheffer matrix S = Stirling1 = (1, log(1x)) = 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
All terms = {1, 3} mod 10.  Muniru A Asiru, Oct 01 2017
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
The above conjecture is true  see the Bala link.  Peter Bala, Jan 20 2018
The terms of this sequence can be derived from the numerators of the fractions, produced by the recursion: b(0) = 1, b(n) = 1 + ((n1) * b(n1)) / (n1 + b(n1)) for n > 0. The denominators give A002720.  Dimitris Valianatos, Aug 01 2018
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


REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 194.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..444 (first 101 terms from T. D. Noe)
A. Aboud, J.P. Bultel, A. Chouria, J.G. Luque, O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 2014.
David Angell, A family of continued fractions, Journal of Number Theory, Volume 130, Issue 4, April 2010, Pages 904911. Section 2.
P. Bala, Integer sequences that become periodic on reduction modulo k for all k
P. Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 4.
P. Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 6.
P. Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 18.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) 11.6.7.
Andreas Bärtschi, Daniel Graf, and Paolo Penna, Truthful Mechanisms for Delivery with Agents, 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems, OASIcs, Volume 59, 2017.
P. Blasiak, A. Horzela, K. A. Penson, G. H. E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffertype polynomials, arXiv:quantph/0501155, 2005.
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quantph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quantph/0402027, 2004.
Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.
J.P. Bultel, A, Chouria, J.G. Luque and O. Mallet, Word symmetric functions and the RedfieldPolya theorem, HAL Id: hal00793788, 2013.
David Callan, Sets, Lists and Noncrossing Partitions , arXiv:0711.4841 [math.CO], 20072008.
David Callan and Emeric Deutsch, The Run Transform, Discrete Math. 312 (2012), no. 19, 29272937, arXiv:1112.3639 [math.CO], 2011.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Clintin P. DavisStober, JeanPaul Doignon, Samuel Fiorini, François Glineur, Michel Regenwetter, Extended Formulations for Order Polytopes through Network Flows, arXiv:1710.02679 [math.OC], 2017.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 125
S. Garrabrant, I. Pak, Words in linear groups, computability and precursiveness, 2015.
Stefan Gerhold, Counting finite languages by total word length, INTEGERS 11 (2011), #A44.
BaiNi Guo and Feng Qi, An Explicit Formula for Bell Numbers in Terms of Stirling Numbers and Hypergeometric Functions, 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 List of predatory publishers).
BaiNi Guo and Feng Qi, An Explicit Formula for Bell Numbers in Terms of Stirling Numbers and Hypergeometric Functions, arXiv:1402.2361 [math.CO], 2014.
T.X. He, A symbolic operator approach to power series transformationexpansion formulas, JIS 11 (2008) 08.2.7
A. Hennessy, P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2
F. Hivert, J.C. Novelli and J.Y. Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 40
Salvador Jacobi, Planning in MultiAgent Systems, Thesis, Technical University of Denmark, Department of Applied Mathematics and Computer Science, 2800 Kongens Lyngby, Denmark, 2014.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; The Mathematica J., 2 (1992), 6778.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 30173023.
Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167176. [Annotated, scanned copy]
JeanChristophe Novelli, JeanYves Thibon, On composition polynomials, arXiv:1510.03033 [math.CO], (11October2015)
OEIS Wiki, Sorting numbers
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinskitype relations via substitution and the moment problem, J. Phys. A 37 (2004), 34753487.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math.CO/0606404, Jan 05 2007.
Feng Qi, On sum of the Lah numbers and zeros of the Kummer confluent hypergeometric function, Research Gate, 2015.
Feng Qi, An Explicit Formula for the Bell Numbers in Terms of the Lah and Stirling Numbers, Mediterranean Journal of Mathematics, November 2015, DOI: 10.1007/s0000901506557.
J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their Nnumbers, not their Anumbers.
Mark A. Shattuck and Carl G. Wagner, Parity Theorems for Statistics on Lattice Paths and Laguerre Configurations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.1.
M. Shattuck, Combinatorial proofs of some Bell number formulas, arXiv preprint arXiv:1401.6588 [math.CO], 2014.
M. Shattuck, Generalized rLah numbers, arXiv preprint arXiv:1412.8721 [math.CO], 2014.
Tomi Silander, Janne Leppäaho, Elias Jääsaari, Teemu Roos, Quotient Normalized Maximum Likelihood Criterion for Learning Bayesian Network Structures, International Conference on Artificial Intelligence and Statistics, 2018.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 8389.
A. Umar, Some combinatorial problems in the theory of symmetric ..., Algebra Disc. Math. 9 (2010) 115126
Thomas Wieder, Further comments on this sequence
Y. Yakubovich, Ergodicity of multiplicative statistics, Journal of Combinatorial Theory, Series A 119 (2012) 12501279, alternative copy.
Index entries for "core" sequences
Index entries for sequences related to Laguerre polynomials
Index entries for related partitioncounting sequences


FORMULA

a(n) = (2*n1)*a(n1)  (n1)*(n2)*a(n2).
E.g.f.: exp( x/(1x) ).
Representation as nth moment of a positive function on positive halfaxis, in Maple notation: a(n) = integral(x^n*exp(x1)*BesselI(1, 2*x^(1/2))/x^(1/2), x =0..infinity), n=1, 2...  Karol A. Penson, Dec 04 2003
a(n) = Sum_{k=0..n} A001263(n, k)*k!.  Philippe Deléham, Dec 10 2003
a(n) = n! Sum_{j=0..n1} binomial(n1, j)/(j+1)!, for n > 0.  Herbert S. Wilf, Jun 14 2005
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
Minor part of this asymptotic expansion is wrong! Right is (in closed form): a(n) ~ n^(n1/4)*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
a(n) = exp(1)*Sum_{m>=0} [m]^n/m!, where [m]^n = m*(m+1)*...*(m+n1) is the rising factorial.  Vladeta Jovovic, Sep 20 2006
Recurrence: D(n,k) = D(n1,k1) + (n1+k) * D(n1,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
Proof: Notice two distinct subsets of the lists for [n]: 1) n is in its own list, then there are D(n1,k1); 2) n is in a list with other numbers. Denoting the separation of lists by , it is not hard to see n has (n1+k) possible positions, so (n1+k) * D(n1,k).  Stephen Dalton (StephenMDalton(AT)gmail.com), Jan 05 2007
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(n1) = exp(1)*f_n(1).  Milan Janjic, May 30 2008
a(n) = (n1)! * Sum_{k=1..n} (a(nk)*k!)/((nk)!*(k1)!), where a(0)=1.  Thomas Wieder, Sep 10 2008
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
From Sergei N. Gladkovskii, Nov 17 2011, Aug 02 2012, Dec 11 2012, Jan 27 2013, Jul 31 2013, Dec 25 2013: (Start)
Continued fractions:
E.g.f.: Q(0) where Q(k) = 1+x/((1x)*(2k+1)x*(1x)*(2k+1)/(x+(1x)*(2k+2)/Q(k+1))).
E.g.f.: 1 + x/(G(0)x) where G(k) = (1x)*k + 1  x*(1x)*(k+1)/G(k+1); (Euler's 1st kind, 1step).
E.g.f.: exp(x/(1x)) = 4/(2(x/(1x))*G(0))1 where G(k) = 1  x^2/(x^2 + 4*(1x)^2*(2*k+1)*(2*k+3)/G(k+1) ).
E.g.f.: 1 + x*(E(0)1)/(x+1) where E(k) = 1 + 1/(k+1)/(1x)/(1x/(x+1/E(k+1) )).
E.g.f.: E(0)/2, where E(k)= 1 + 1/(1  x/(x + (k+1)*(1x)/E(k+1) )).
E.g.f.: E(0)1, where E(k) = 2 + x/( (2*k+1)*(1x)  x/E(k+1) ).
(End)
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
a(n) = n!*hypergeom([1n],[2],1) for n >= 1.  Peter Luschny, Jun 05 2014
a(n) = (1)^(n1)*KummerU(1n,2,1).  Peter Luschny, Sep 17 2014
a(n) = hypergeom([n+1, n], [], 1) for n >= 0.  Peter Luschny, Apr 08 2015
E.g.f.: Product_{k>0} exp(x^k).  Franklin T. AdamsWatters, May 11 2016
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
G.f. G(x)=y satisfies a differential equation: (1x)*y2*(1x)*x^2*y'+x^4*y''=1.  Bradley Klee, Aug 13 2018


EXAMPLE

Illustration of first terms as sets of ordered lists of the first n integers:
a(1) = 1 : (1)
a(2) = 3 : (12), (21), (1)(2).
a(3) = 13 : (123) (6 ways), (12)(3) (2*3 ways) (1)(2)(3) (1 way)
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).
:
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 + ...


MAPLE

a := proc(n) option remember: if n=0 then RETURN(1) fi: if n=1 then RETURN(1) fi: (2*n1)*a(n1)  (n1)*(n2)*a(n2) end:for n from 0 to 20 do printf(`%d, `, a(n)) od:
spec := [S, {S = Set(Prod(Z, Sequence(Z)))}, labeled]; [seq(combstruct[count](spec, size=n), n=0..40)];
with(combinat):seq(sum(abs(stirling1(n, k))*bell(k), k=0..n), n=0..18); # Zerinvary Lajos, Nov 26 2006
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
a := n > n!*hypergeom([1  n], [2], 1): seq(round(evalf(a(n), 32)), n=0..19); # Peter Luschny, Jun 05 2014


MATHEMATICA

Range[0, 19]! CoefficientList[ Series[E^(x/(1  x)), {x, 0, 19}], x] (* Robert G. Wilson v, Apr 04 2005 *)
a[ n_] := If[ n<0, 0, n! SeriesCoefficient[ Exp[ x / (1  x)], {x, 0, n}]]; (* Michael Somos, Jul 19 2005 *)
a[n_]:=If[n==0, 1, n! Sum[Binomial[n1, j]/(j+1)!, {j, 0, n1}]]; Table[a[n], {n, 0, 30}] (* Wilf *)
a[0] = 1; a[n_] := n!*Hypergeometric1F1[n+1, 2, 1]/E; Table[a[n], {n, 0, 19}] (* JeanFrançois Alcover, Jun 18 2012, after Shai Covo *)
Table[Sum[BellY[n, k, Range[n]!], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)


PROG

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( x / (1  x) + x * O(x^n)), n))}; /* Michael Somos, Feb 10 2005 */
(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 */
(PARI) {a(n) = s = 1; for(k = 1, n1, s = 1 + k * s / (k + s)); return( numerator(s))}; /* Dimitris Valianatos, Aug 03 2018 */
(Maxima) makelist(sum(abs(stirling1(n, k))*belln(k), k, 0, n), n, 0, 24); /* Emanuele Munarini, Jul 04 2011 */
(Maxima) makelist(hypergeometric([n+1, n], [], 1), n, 0, 12); /* Emanuele Munarini, Sep 27 2016 */
(Haskell)
a000262 n = a000262_list !! n
a000262_list = 1 : 1 : zipWith ()
(tail $ zipWith (*) a005408_list a000262_list)
(zipWith (*) a002378_list a000262_list)
 Reinhard Zumkeller, Mar 06 2014
(Sage)
A000262 = lambda n: hypergeometric([n+1, n], [], 1)
[simplify(A000262(n)) for n in (0..19)] # Peter Luschny, Apr 08 2015
(GAP)
a:=[1, 1];; for n in [3..10^2] do a[n]:=(2*n3)*a[n1](n2)*(n3)*a[n2]; od; A000262:=a; # Muniru A Asiru, Oct 01 2017


CROSSREFS

a(n), n >= 1, is sum of nth row of A008297 (unsigned Lahtriangle).  Wolfdieter Lang
A002868 = maximal element of nth row of A008297.
Cf. A001263, A001700, A002378, A005408, A066668.
Cf. A111596 (unsigned row sums of triangle).
Cf. A052852.  David Angell (angell(AT)maths.unsw.edu.au), Dec 18 2008
Main diagonal of A257740 and of A319501.
Cf. A000110, A132393, A082579, A255807, A255819, A318976.
Sequence in context: A193932 A193933 A293125 * A318617 A059294 A124468
Adjacent sequences: A000259 A000260 A000261 * A000263 A000264 A000265


KEYWORD

nonn,easy,core,nice,changed


AUTHOR

N. J. A. Sloane


STATUS

approved



