A052852 -id:A052852 - cp's OEIS frontend

A000262 Number of "sets of lists": number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset.

1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553, 58941091, 824073141, 12470162233, 202976401213, 3535017524403, 65573803186921, 1290434218669921, 26846616451246353, 588633468315403843, 13564373693588558173, 327697927886085654441, 8281153039765859726341

Offset: 0

N. J. A. Sloane

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
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
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
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/(1-x)]. - 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, 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
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
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
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
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
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
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
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
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
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 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
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
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 + ((n-1) * b(n-1)) / (n-1 + b(n-1)) 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
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
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

			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 + ...

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).

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Index entries for "core" sequences
Index entries for sequences related to Laguerre polynomials
Index entries for related partition-counting sequences

Row sums of A271703 and for n >= 1 of A008297. Unsigned row sums of A111596.
A002868 is maximal element of the n-th row of A271703 and for n >= 1 of A008297.
Main diagonal of A257740 and of A319501.
Cf. A001263, A001700, A002378, A005408, A066668, A056953, A002720.
Cf. A000110, A052852, A082579, A132393, A255807, A255819, A318976.

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

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

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

[Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), -1): n in [0..30]]; // G. C. Greubel, Feb 23 2021

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:
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 -> `if`(n=0, 1, n!*hypergeom([1 - n], [2], -1)): seq(simplify(a(n)), n=0..19); # Peter Luschny, Jun 05 2014

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[n-1, j]/(j+1)!, {j, 0, n-1}]]; 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}] (* Jean-François 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 *)
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 *)
Table[n!*LaguerreL[n, -1, -1], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *)
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 *)

makelist(sum(abs(stirling1(n,k))*belln(k),k,0,n),n,0,24); /* Emanuele Munarini, Jul 04 2011 */

makelist(hypergeometric([-n+1,-n],[],1),n,0,12); /* Emanuele Munarini, Sep 27 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp( x / (1 - x) + x * O(x^n)), n))}; /* Michael Somos, Feb 10 2005 */

{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 */

{a(n) = s = 1; for(k = 1, n-1, s = 1 + k * s / (k + s)); return( numerator(s))}; /* Dimitris Valianatos, Aug 03 2018 */

{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 */

a(n) = if (n==0, 1, (n-1)!*pollaguerre(n-1,1,-1)); \\ Michel Marcus, Feb 23 2021; Jul 13 2024

from sympy import hyper, hyperexpand
def A000262(n): return hyperexpand(hyper((-n+1, -n), [], 1)) # Chai Wah Wu, Jan 14 2022

A000262 = lambda n: hypergeometric([-n+1, -n], [], 1)
[simplify(A000262(n)) for n in (0..19)] # Peter Luschny, Apr 08 2015

D-finite with recurrence: a(n) = (2*n-1)*a(n-1) - (n-1)*(n-2)*a(n-2).
E.g.f.: exp( x/(1-x) ).
a(n) = Sum_{k=0..n} |A008275(n,k)| * A000110(k). - Vladeta Jovovic, Feb 01 2003
a(n) = (n-1)!*LaguerreL(n-1,1,-1) for n >= 1. - Vladeta Jovovic, May 10 2003
Representation as n-th moment of a positive function on positive half-axis: a(n) = Integral_{x=0..oo} x^n*exp(-x-1)*BesselI(1, 2*x^(1/2))/x^(1/2) dx, n >= 1. - 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..n-1} binomial(n-1, 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^(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
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
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
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
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
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
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
a(n) = n!* A067764(n) / A067653(n). - Gary Detlefs, May 15 2010
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/((1-x)*(2k+1)-x*(1-x)*(2k+1)/(x+(1-x)*(2k+2)/Q(k+1))).
E.g.f.: 1 + x/(G(0)-x) where G(k) = (1-x)*k + 1 - x*(1-x)*(k+1)/G(k+1).
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) ).
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) )).
E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - x/(x + (k+1)*(1-x)/E(k+1) )).
E.g.f.: E(0)-1, where E(k) = 2 + x/( (2*k+1)*(1-x) - 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([1-n],[2],-1) for n >= 1. - Peter Luschny, Jun 05 2014
a(n) = (-1)^(n-1)*KummerU(1-n,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. Adams-Watters, 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: (1-x)*y-2*(1-x)*x^2*y'+x^4*y''=1. - Bradley Klee, Aug 13 2018
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
3 divides a(3*n-1); 9 divides a(9*n-1); 11 divides a(11*n-1). - Peter Bala, Mar 26 2022
For n > 0, a(n) = Sum_{k=0..n-1}*k!*C(n-1,k)*C(n,k). - Francesca Aicardi, Nov 03 2022
For n > 0, a(n) = (n-1)! * (Sum_{i=0..n-1} A002720(i) / i!). - Werner Schulte, Mar 29 2024
a(n+1) = numerator of (1 + n/(1 + n/(1 + (n-1)/(1 + (n-1)/(1 + ... + 1/(1 + 1/(1))))))). See A002720 for the denominators. - Peter Bala, Feb 11 2025

A005493 2-Bell numbers: a(n) = number of partitions of [n+1] with a distinguished block.

Original entry on oeis.org

1, 3, 10, 37, 151, 674, 3263, 17007, 94828, 562595, 3535027, 23430840, 163254885, 1192059223, 9097183602, 72384727657, 599211936355, 5150665398898, 45891416030315, 423145657921379, 4031845922290572, 39645290116637023, 401806863439720943, 4192631462935194064

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of Boolean sublattices of the Boolean lattice of subsets of {1..n}.
a(n) = p(n+1) where p(x) is the unique degree n polynomial such that p(k) = A000110(k+1) for k = 0, 1, ..., n. - Michael Somos, Oct 07 2003
With offset 1, number of permutations beginning with 12 and avoiding 21-3.
Rows sums of Bell's triangle (A011971). - Jorge Coveiro, Dec 26 2004
Number of blocks in all set partitions of an (n+1)-set. Example: a(2)=10 because the set partitions of {1,2,3} are 1|2|3, 1|23, 12|3, 13|2 and 123, with a total of 10 blocks. - Emeric Deutsch, Nov 13 2006
Number of partitions of n+3 with at least one singleton and with the smallest element in a singleton equal to 2. - Olivier Gérard, Oct 29 2007
See page 29, Theorem 5.6 of my paper on the arXiv: These numbers are the dimensions of the homogeneous components of the operad called ComTrip associated with commutative triplicial algebras. (Triplicial algebras are related to even trees and also to L-algebras, see A006013.) - Philippe Leroux, Nov 17 2007
Number of set partitions of (n+2) elements where two specific elements are clustered separately. Example: a(1)=3 because 1/2/3, 1/23, 13/2 are the 3 set partitions with 1, 2 clustered separately. - Andrey Goder (andy.goder(AT)gmail.com), Dec 17 2007
Equals A008277 * [1,2,3,...], i.e., the product of the Stirling number of the second kind triangle and the natural number vector. a(n+1) = row sums of triangle A137650. - Gary W. Adamson, Jan 31 2008
Prefaced with a "1" = row sums of triangle A152433. - Gary W. Adamson, Dec 04 2008
Equals row sums of triangle A159573. - Gary W. Adamson, Apr 16 2009
Number of embedded coalitions in an (n+1)-person game. - David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008
If prefixed with 0, gives first differences of Bell numbers A000110 (cf. A106436). - N. J. A. Sloane, Aug 29 2013
Sum_{n>=0} a(n)/n! = e^(e+1) = 41.19355567... (see A235214). Contrast with e^(e-1) = Sum_{n>=0} A000110(n)/n!. - Richard R. Forberg, Jan 05 2014

			For example, a(1) counts (12), (1)-2, 1-(2) where dashes separate blocks and the distinguished block is parenthesized.

Olivier Gérard and Karol A. Penson, A budget of set partition statistics, in preparation. Unpublished as of 2017.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Chai Wah Wu, Table of n, a(n) for n = 0..574 (first 101 terms from T. D. Noe)
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
M. Brookes, J. East, C. Miller, J.D. Mitchell, and N. Ruskuc, Heights of one- and two-sided congruence lattices of semigroups, arXiv:2310.08229 [math.GR], 2023.
Giulio Cerbai, Modified ascent sequences and Bell numbers, arXiv:2305.10820 [math.CO], 2023. See p. 11.
Martin Cohn, Shimon Even, Karl Menger, Jr., and Philip K. Hooper, On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
Martin Cohn, Shimon Even, Karl Menger, Jr., and Philip K. Hooper, On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
R. B. Corcino and C. B. Corcino, On generalized Bell polynomials, Discrete Dyn. Nat. Soc. Article ID: 623456 (2011).
C. B. Corcino and R. B. Corcino, An asymptotic formula for r-Bell numbers with real arguments, ISRN Discrete Math, 2013 (2013), Article ID 274697.
Gábor Czédli, Lattices with lots of congruence energy, arXiv:2205.02294 [math.RA], 2022.
A. Dil and V. Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series II, Appl. Anal. Discrete Math. 5 (2011), 212-229.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
A. L. L. Gao, S. Kitaev, and P. B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.
S. K. Ghosal and J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
Alain Hertz, Hadrien Mélot, Sébastien Bonte, Gauvain Devillez, and Pierre Hauweele, Upper bounds on the average number of colors in the non-equivalent colorings of a graph, arXiv:2105.01120 [math.CO], 2021.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 152
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).
S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Philippe Leroux, L-algebras, triplicial-algebras, within an equivalence of categories motivated by graphs (formerly "An equivalence of categories motivated by weighted directed graphs"), arXiv:0709.3453 [math-ph], 2007-2008.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
I. Mezo, The r-Bell numbers, J. Integer Seq. 14(1) (2011), Article 11.1.1.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 12.16 (pdf, ps)
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
J. Riordan, Letter, Oct 31 1977
Eric Weisstein's World of Mathematics, Stirling Transform.
Eric Weisstein's World of Mathematics, Bell Triangle
E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.
David W. K. Yeung, Recursive sequence identifying the number of embedded coalitions, International Game Theory Review 10 (1) (2008), 129-136.

Cf. A000110, A005494, A008277, A011971, A011968, A049020, A106436, A124323, A137650, A152433, A159573.
A row or column of the array A108087.
Row sums of triangle A143494. - Wolfdieter Lang, Sep 29 2011. And also of triangle A362924. - N. J. A. Sloane, Aug 10 2023

with(combinat): seq(bell(n+2)-bell(n+1),n=0..22); # Emeric Deutsch, Nov 13 2006
seq(add(binomial(n, k)*(bell(n-k)), k=1..n), n=1..23); # Zerinvary Lajos, Dec 01 2006
A005493  := proc(n) local a,b,i;
a := [seq(3,i=1..n)]; b := [seq(2,i=1..n)];
2^n*exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=1,%),66)) end:
seq(A005493(n),n=0..22); # Peter Luschny, Mar 30 2011
BT := proc(n,k) option remember; if n = 0 and k = 0 then 1
elif k = n then BT(n-1,0) else BT(n,k+1)+BT(n-1,k) fi end:
A005493 := n -> add(BT(n,k),k=0..n):
seq(A005493(i),i=0..22); # Peter Luschny, Aug 04 2011
# For Maple code for r-Bell numbers, etc., see A232472. - N. J. A. Sloane, Nov 27 2013

a=Exp[x]-1; Rest[CoefficientList[Series[a Exp[a],{x,0,20}],x] * Table[n!,{n,0,20}]]
a[ n_] := If[ n<0, 0, With[ {m = n+1}, m! SeriesCoefficient[ # Exp@# &[ Exp@x - 1], {x, 0, m}]]]; (* Michael Somos, Nov 16 2011 *)
Differences[BellB[Range[30]]] (* Harvey P. Dale, Oct 16 2014 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) + 2*x - 1), n))}; /* Michael Somos, Oct 09 2002 */

{a(n) = if( n<0, 0, n+=2; subst( polinterpolate( Vec( serlaplace( exp( exp( x + O(x^n)) - 1) - 1))), x, n))}; /* Michael Somos, Oct 07 2003 */

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A005493_list, blist, b = [], [1], 1
for _ in range(1001):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A005493_list.append(blist[-2])
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

a(n-1) = Sum_{k=1..n} k*Stirling2(n, k) for n>=1.
E.g.f.: exp(exp(x) + 2*x - 1). First differences of Bell numbers (if offset 1). - Michael Somos, Oct 09 2002
G.f.: Sum_{k>=0} (x^k/Product_{l=1..k} (1-(l+1)x)). - Ralf Stephan, Apr 18 2004
a(n) = Sum_{i=0..n} 2^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007 [Written umbrally, a(n) = (B+2)^n. - N. J. A. Sloane, Feb 07 2009]
Representation as an infinite series: a(n-1) = Sum_{k>=2} (k^n*(k-1)/k!)/exp(1), n=1, 2, ... This is a Dobinski-type summation formula. - Karol A. Penson, Mar 14 2002
Row sums of A011971 (Aitken's array, also called Bell triangle). - Philippe Deléham, Nov 15 2003
a(n) = exp(-1)*Sum_{k>=0} ((k+2)^n)/k!. - Gerald McGarvey, Jun 03 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} (1+binomial(n, j))*a(j). - Jon Perry, Apr 25 2005
a(n) = A000296(n+3) - A000296(n+1). - Philippe Deléham, Jul 31 2005
a(n) = B(n+2) - B(n+1), where B(n) are Bell numbers (A000110). - Franklin T. Adams-Watters, Jul 13 2006
a(n) = A123158(n,2). - Philippe Deléham, Oct 06 2006
Binomial transform of Bell numbers 1, 2, 5, 15, 52, 203, 877, 4140, ... (see A000110).
Define f_1(x), f_2(x), ... such that f_1(x)=x*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 2008
Representation of numbers a(n), n=0,1..., as special values of hypergeometric function of type (n)F(n), in Maple notation: a(n)=exp(-1)*2^n*hypergeom([3,3...3],[2.2...2],1), n=0,1..., i.e., having n parameters all equal to 3 in the numerator, having n parameters all equal to 2 in the denominator and the value of the argument equal to 1. Examples: a(0)= 2^0*evalf(hypergeom([],[],1)/exp(1))=1 a(1)= 2^1*evalf(hypergeom([3],[2],1)/exp(1))=3 a(2)= 2^2*evalf(hypergeom([3,3],[2,2],1)/exp(1))=10 a(3)= 2^3*evalf(hypergeom([3,3,3],[2,2,2],1)/exp(1))=37 a(4)= 2^4*evalf(hypergeom([3,3,3,3],[2,2,2,2],1)/exp(1))=151 a(5)= 2^5*evalf(hypergeom([3,3,3,3,3],[2,2,2,2,2],1)/exp(1)) = 674. - Karol A. Penson, Sep 28 2007
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) = (-1)^(n)charpoly(A,-2). - Milan Janjic, Jul 08 2010
a(n) = D^(n+1)(x*exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A003128, A052852 and A009737. - Peter Bala, Nov 25 2011
From Sergei N. Gladkovskii, Oct 11 2012 to Jan 26 2014: (Start)
Continued fractions:
G.f.: 1/U(0) where U(k) = 1 - x*(k+3) - x^2*(k+1)/U(k+1).
G.f.: 1/(U(0)-x) where U(k) = 1 - x - x*(k+1)/(1 - x/U(k+1)).
G.f.: G(0)/(1-x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k+2*x-1) - x*(2*k+1)*(2*k+3)*(2*x*k+2*x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+3*x-1)/G(k+1) )).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-2*x-k*x)/(1-x/(x-1/G(k+1) )).
G.f.: -G(0)/x where G(k) = 1 - 1/(1-k*x-x)/(1-x/(x-1/G(k+1) )).
G.f.: 1 - 2/x + (1/x-1)*S where S = sum(k>=0, ( 1 + (1-x)/(1-x-x*k) )*(x/(1-x))^k / prod(i=0..k-1, (1-x-x*i)/(1-x) ) ).
G.f.: (1-x)/x/(G(0)-x) - 1/x where G(k) = 1 - x*(k+1)/(1 - x/G(k+1) ).
G.f.: (1/G(0) - 1)/x^3 where G(k) = 1 - x/(x - 1/(1 + 1/(x*k-1)/G(k+1) )).
G.f.: 1/Q(0), where Q(k)= 1 - 2*x - x/(1 - x*(k+1)/Q(k+1)).
G.f.: G(0)/(1-3*x), where G(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1 - x*(k+3))*(1 - x*(k+4))/G(k+1) ). (End)
a(n) ~ exp(n/LambertW(n) + 3*LambertW(n)/2 - n - 1) * n^(n + 1/2) / LambertW(n)^(n+1). - Vaclav Kotesovec, Jun 09 2020
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) ~ n^2 * Bell(n) / LambertW(n)^2 * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021
a(n) = Sum_{k=0..n} 3^k*A124323(n, k). - Mélika Tebni, Jun 02 2022

Definition revised by David Callan, Oct 11 2005

A062139 Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x).

Original entry on oeis.org

1, 3, -1, 12, -8, 1, 60, -60, 15, -1, 360, -480, 180, -24, 1, 2520, -4200, 2100, -420, 35, -1, 20160, -40320, 25200, -6720, 840, -48, 1, 181440, -423360, 317520, -105840, 17640, -1512, 63, -1, 1814400, -4838400, 4233600

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,2,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^3. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} |A008297(n,m)|*(-x)^m, n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
This unsigned matrix is embedded in the matrix for n!*L(n,-2,-x). Introduce 0,0 to each unsigned row and then add 1,-1,1 to the array as the first two rows to generate n!*L(n,-2,-x). - Tom Copeland, Apr 20 2014
The unsigned n-th row reverse polynomial equals the numerator polynomial of the finite continued fraction 1 - x/(1 + (n+1)*x/(1 + n*x/(1 + n*x/(1 + ... + 2*x/(1 + 2*x/(1 + x/(1 + x/(1)))))))). Cf. A089231. The denominator polynomial of the continued fraction is the (n+1)-th row polynomial of A144084. An example is given below. - Peter Bala, Oct 06 2019

			Triangle begins:
     1;
     3,    -1;
    12,    -8,    1;
    60,   -60,   15,   -1;
   360,  -480,  180,  -24,  1;
  2520, -4200, 2100, -420, 35, -1;
  ...
2!*L(2,2,x) = 12 - 8*x + x^2.
Unsigned row 3 polynomial in reverse form as the numerator of a continued fraction: 1 - x/(1 + 4*x/(1 + 3*x/(1 + 3*x/(1 + 2*x/(1 + 2*x/(1 + x/(1 + x))))))) = (60*x^3 + 60*x^2 + 15*x + 1)/(24*x^4 + 96*x^3 + 72*x^2 + 16*x + 1). - _Peter Bala_, Oct 06 2019

Indranil Ghosh, Rows 0..125, flattened
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001710, A005990, A005461, A062193-A062195. The row sums (signed) give A062197, the row sums (unsigned) give A052852.

Cf. A021009, A062137-A062140, A066667, A089231, A144084.

with(PolynomialTools):
p := n -> (n+2)!*hypergeom([-n],[3],x)/2:
seq(CoefficientList(simplify(p(n)), x), n=0..9); # Peter Luschny, Apr 08 2015

Flatten[Table[((-1)^m)*n!*Binomial[n+2,n-m]/m!,{n,0,8},{m,0,n}]] (* Indranil Ghosh, Feb 24 2017 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(((-1)^k)*n!*binomial(n+2, n-k)/k!, ", ");); print(););} \\ Michel Marcus, May 06 2014

row(n) = Vecrev(n!*pollaguerre(n, 2)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n,r):return f(n)//f(r)//f(n-r)
i=0
for n in range(16):
    for m in range(n+1):
        i += 1
        print(i,((-1)**m)*f(n)*C(n+2,n-m)//f(m)) # Indranil Ghosh, Feb 24 2017

from functools import cache
@cache
def T(n, k):
    if k < 0 or k > n: return 0
    if k == n: return (-1)**n
    return (n + k + 2) * T(n-1, k) - T(n-1, k-1)
for n in range(7): print([T(n,k) for k in range(n + 1)])
# Peter Luschny, Mar 25 2024

T(n, m) = ((-1)^m)*n!*binomial(n+2, n-m)/m!.
E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^3), m >= 0.
n!*L(n,2,x) = (n+2)!*hypergeom([-n],[3],x)/2. - Peter Luschny, Apr 08 2015
From Werner Schulte, Mar 24 2024: (Start)
T(n, k) = (n+k+2) * T(n-1, k) - T(n-1, k-1) with initial values T(0, 0) = 1 and T(i, j) = 0 if j < 0 or j > i.
T = T^(-1), i.e., T is matrix inverse of T. (End)

A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.

Original entry on oeis.org

2, 3, 7, 4, 13, 34, 5, 21, 73, 209, 6, 31, 136, 501, 1546, 7, 43, 229, 1045, 4051, 13327, 8, 57, 358, 1961, 9276, 37633, 130922, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114, 11, 111, 1021, 8501

Offset: 1

Hugo Pfoertner, Aug 22 2003

Compare with A088699. - Peter Bala, Sep 17 2008

T(m, n) gives the number of matchings in the complete bipartite graph K_{m,n}. - Eric W. Weisstein, Apr 25 2017

			One person:
T(1,1)=a(1)=2: 0,1 (seat empty or occupied);
T(2,1)=a(2)=3: 00,10,01 (both seats empty, left seat occupied, right seat occupied).
Two persons:
T(2,2)=a(3)=7: 00,10,01,20,02,12,21;
T(3,2)=a(5)=13: 000,100,010,001,200,020,002,120,102,012,210,201,021.
Triangle starts:
  2;
  3  7;
  4 13  34;
  5 21  73 209;
  6 31 136 501 1546;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)
Ed Jones, Number of seatings, discussion in newsgroup sci.math, Aug 9, 2003.
R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or columns, Table 1.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Luca Zecchini, Tobias Bleifuß, Giovanni Simonini, Sonia Bergamaschi, and Felix Naumann, Determining the Largest Overlap between Tables, Proc. ACM Manag. Data (SIGMOD 2024) Vol. 2, No. 1, Art. 48. See p. 48:6.
Index entries for sequences related to Laguerre polynomials

Diagonal: A002720, first subdiagonal: A000262, 2nd subdiagonal: A052852, 3rd subdiagonal: A062147, 4th subdiagonal: A062266, 5th subdiagonal: A062192, 2nd row/column: A002061. With column 0: A176120.

[Factorial(k)*Evaluate(LaguerrePolynomial(k, n-k), -1): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021

A086885 := proc(n,k)
    add( binomial(n,j)*binomial(k,j)*j!,j=0..min(n,k)) ;
end proc: # R. J. Mathar, Dec 19 2014

Table[Table[Sum[k! Binomial[n, k] Binomial[j, k], {k, 0, j}], {j, 1, n}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Jul 09 2015 *)
Table[m! LaguerreL[m, n - m, -1], {n, 10}, {m, n}] // Flatten (* Eric W. Weisstein, Apr 25 2017 *)

T(i, j) = j!*pollaguerre(j, i-j, -1); \\ Michel Marcus, Feb 23 2021

flatten([[factorial(k)*gen_laguerre(k, n-k, -1) for k in [1..n]] for n in (1..10)]) # G. C. Greubel, Feb 23 2021

a(n) = T(i, j) with n=(i*(i-1))/2+j; T(i, 1)=i+1, T(i, j)=T(i, j-1)+i*T(i-1, j-1) for j>1.
The role of seats and persons may be interchanged, so T(i, j)=T(j, i).
T(i, j) = j!*LaguerreL(j, i-j, -1). - Vladeta Jovovic, Aug 25 2003
T(i, j) = Sum_{k=0..j} k!*binomial(i, k)*binomial(j, k). - Vladeta Jovovic, Aug 25 2003

A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 13, 21, 9, 1, 73, 136, 78, 16, 1, 501, 1045, 730, 210, 25, 1, 4051, 9276, 7515, 2720, 465, 36, 1, 37633, 93289, 85071, 36575, 8015, 903, 49, 1, 394353, 1047376, 1053724, 519456, 137270, 20048, 1596, 64, 1, 4596553, 12975561

Offset: 0

Vladeta Jovovic, Jan 04 2001

L'(n,i) are unsigned Lah numbers (cf. A008297): L'(n,i)=n!/i!*binomial(n-1,i-1) for i >= 1, L'(0,0)=1, L'(n,0)=0 for n>0. T(n,0)=A000262(n); T(n,2)=A052852(n). Row sums A052897.
Exponential Riordan array [e^(x/(1-x)),x/(1-x)]. - Paul Barry, Apr 28 2007
From Wolfdieter Lang, Jun 22 2017: (Start)
The inverse matrix T^(-1) is exponential Riordan (aka Sheffer) (e^(-x), x/(1+x)): T^(-1)(n, m) = (-1)^(n-m)*A271705(n, m).
The a- and z-sequences of this Sheffer (aka exponential Riordan) matrix are a = [1,1,repeat(0)] and z(n) = (-1)^(n+1)*A028310(n)/A000027(n-1) with e.g.f. ((1+x)/x)*(1-exp(-x)). For a- and z-sequences see a W. Lang link under A006232 with references. (End)

			The triangle T = A007318*A271703 starts:
n\m       0        1        2       3       4      5     6    7  8 9 ...
0:        1
1:        1        1
2:        3        4        1
3:       13       21        9       1
4:       73      136       78      16       1
5:      501     1045      730     210      25      1
6:     4051     9276     7515    2720     465     36     1
7:    37633    93289    85071   36575    8015    903    49    1
8:   394353  1047376  1053724  519456  137270  20048  1596   64  1
9:  4596553 12975561 14196708 7836276 2404206 427518 44436 2628 81 1
... reformatted. - _Wolfdieter Lang_, Jun 22 2017
E.g.f. for T(n, 2) = 1/2!*(x/(1-x))^2*e^(x/(x-1)) = 1*x^2/2 + 9*x^3/3! + 78*x^4/4! + 730*x^5/5! + 7515*x^6/6 + ...
From _Wolfdieter Lang_, Jun 22 2017: (Start)
The z-sequence starts: [1, 1/2, -2/3, 3/4, -4/5, 5/6, -6/7, 7/8, -8/9, ...
T recurrence: T(3, 0) = 3*(1*T(2,0) + (1/2)*T(2, 1) + (-2/3)*T(2 ,1)) = 3*(3 + (1/2)*4 - (2/3)) = 13; T(3, 1) = 3*(T(2, 0)/1 + T(2, 1)) = 3*(3 + 4) = 21.
Meixner type recurrence for R(2, x): (D - D^2)*(3 + 4*x + x^2) = 4 + 2*x - 2 = 2*(1 + x), (D = d/dx).
General Sheffer recurrence for R(2, x): (1+x)*(1 + 2*D + D^2)*(1 + x) = (1+x)*(1 + x + 2) = 3 + 4*x + x^2. (End)

Muniru A Asiru, Rows n=0..50 of triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A007318, A008297, A052852, A052897, A271703, A271705.

Concatenation([1],Flat(List([1..10],n->List([0..n],m->Sum([0..n],i-> Factorial(n)/Factorial(i)*Binomial(n-1,i-1)*Binomial(i,m)))))); # Muniru A Asiru, Jul 25 2018

A059110:= func< n,k | n eq 0 select 1 else Factorial(n-1)*Binomial(n,k)*Evaluate(LaguerrePolynomial(n-1, 1-k), -1) >;
[A059110(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Lprime := proc(n,i)
    if n = 0 and i = 0 then
        1;
    elif k = 0 then
        0 ;
    else
        n!/i!*binomial(n-1,i-1) ;
    end if;
end proc:
A059110 := proc(n,k)
    add(Lprime(n,i)*binomial(i,k),i=0..n) ;
end proc: # R. J. Mathar, Mar 15 2013

(* First program *)
lp[n_, i_] := Binomial[n-1, i-1]*n!/i!; lp[0, 0] = 1; t[n_, m_] := Sum[lp[n, i]*Binomial[i, m], {i, 0, n}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)
(* Second program *)
A059110[n_, k_]:= If[n==0, 1, (n-1)!*Binomial[n, k]*LaguerreL[n-1, 1-k, -1]];
Table[A059110[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

def A059110(n, k): return 1 if n==0 else factorial(n-1)*binomial(n, k)*gen_laguerre(n-1, 1-k, -1)
flatten([[A059110(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

E.g.f. for column m: (1/m!)*(x/(1-x))^m*e^(x/(x-1)), m >= 0.
From Wolfdieter Lang, Jun 22 2017: (Start)
E.g.f. for row polynomials in powers of x (e.g.f. of the triangle): exp(z/(1-z))* exp(x*z/(1-z)) (exponential Riordan).
Recurrence: T(n, 0) = Sum_{j=0} z(j)*T(n-1, j), n >= 1, with z(n) = (-1)^(n+1)*A028310(n), T(0, 0) = 1, T(n, m) = 0 n < m, T(n, m) = n*(T(n-1, m-1)/m + T(n-1, m)), n >= m >= 1 (from the z- and a-sequence, see a comment above).
Meixner type recurrence for the (monic) row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: Sum_{k=0..n-1} (-1)^k*D^(k+1)*R(n, x) = n*R(n-1, x), n >=1, R(0, x) = 1, with D = d/dx.
General Sheffer recurrence: R(n, x) = (x+1)*(1+D)^2*R(n-1, x), n >=1, R(0, x) = 1.
(End)
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of this entry; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 18 2018
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (n-1)!*binomial(n, k)*LaguerreL(n-1, 1-k, -1) with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A052897(n). (End)

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A000262 Number of "sets of lists": number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset.

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A005493 2-Bell numbers: a(n) = number of partitions of [n+1] with a distinguished block.

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A062139 Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x).

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A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.

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A059110 Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.

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A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

A123510 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

A123511 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.

A123512 Arises in the normal ordering of functions of a(a+)a, where a and a+ are the boson annihilation and creation operators, respectively.