A271703 -id:A271703 - 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

A008297 Triangle of Lah numbers.

Original entry on oeis.org

-1, 2, 1, -6, -6, -1, 24, 36, 12, 1, -120, -240, -120, -20, -1, 720, 1800, 1200, 300, 30, 1, -5040, -15120, -12600, -4200, -630, -42, -1, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, -362880, -1451520, -1693440, -846720, -211680, -28224, -2016, -72, -1, 3628800, 16329600, 21772800, 12700800

Offset: 1

N. J. A. Sloane

|a(n,k)| = number of partitions of {1..n} into k lists, where a list means an ordered subset.
Let N be a Poisson random variable with parameter (mean) lambda, and Y_1,Y_2,... independent exponential(theta) variables, independent of N, so that their density is given by (1/theta)*exp(-x/theta), x > 0. Set S=Sum_{i=1..N} Y_i. Then E(S^n), i.e., the n-th moment of S, is given by (theta^n) * L_n(lambda), n >= 0, where L_n(y) is the Lah polynomial Sum_{k=0..n} |a(n,k)| * y^k. - Shai Covo (green355(AT)netvision.net.il), Feb 09 2010
For y = lambda > 0, formula 2) for the Lah polynomial L_n(y) dated Feb 02 2010 can be restated as follows: L_n(lambda) is the n-th ascending factorial moment of the Poisson distribution with parameter (mean) lambda. - Shai Covo (green355(AT)netvision.net.il), Feb 10 2010
See A111596 for an expression of the row polynomials in terms of an umbral composition of the Bell polynomials and relation to an inverse Mellin transform and a generalized Dobinski formula. - Tom Copeland, Nov 21 2011
Also the Bell transform of the sequence (-1)^(n+1)*(n+1)! without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016
Named after the Slovenian mathematician and actuary Ivo Lah (1896-1979). - Amiram Eldar, Jun 13 2021

			|a(2,1)| = 2: (12), (21); |a(2,2)| = 1: (1)(2). |a(4,1)| = 24: (1234) (24 ways); |a(4,2)| = 36: (123)(4) (6*4 ways), (12)(34) (3*4 ways); |a(4,3)| = 12: (12)(3)(4) (6*2 ways); |a(4,4)| = 1: (1)(2)(3)(4) (1 way).
Triangle:
    -1;
     2,    1;
    -6,   -6,   -1;
    24,   36,   12,   1;
  -120, -240, -120, -20, -1; ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
Shai Covo, The moments of a compound Poisson process with exponential or centered normal jumps, J. Probab. Stat. Sci., Vol. 7, No. 1 (2009), pp. 91-100.
Theodore S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176; the sequence called {!}^{n+}. For a link to this paper see A000262.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
S. Gill Williamson, Combinatorics for Computer Science, Computer Science Press, 1985; see p. 176.

T. D. Noe, Rows n=1..100 of triangle, flattened
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
P. Blasiak, K. A. Penson, and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
Tom Copeland, Lagrange a la Lah, 2011.
Siad Daboul, Jan Mangaldan, Michael Z. Spivey, and Peter J. Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., Vol. 86, No. 1 (2013), pp. 39-47.
Askar Dzhumadil'daev and Damir Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014-2015. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, Vol. 22, No. 4 (2015), #P4.10.
B. S. El-Desouky, Nenad P.Cakić, and Toufik Mansour, Modified approach to generalized Stirling numbers via differential operators, Appl. Math. Lett., Vol. 23, No. 1 (2010), pp. 115-120.
Sen-Peng Eu, Tung-Shan Fu, Yu-Chang Liang, and Tsai-Lien Wong. On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks. arXiv:1701.00600 [math.CO], 2017.
Milan Janjic, Some classes of numbers and derivatives, JIS, Vol. 12 (2009), pp. 09.8.3
Dmitry Karp and Elena Prilepkina, Generalized Stieltjes transforms: basic aspects, arXiv preprint arXiv:1111.4271 [math.CA], 2011.
Dmitry Karp and Elena Prilepkina, Generalized Stieltjes functions and their exact order, Journal of Classical Analysis, Vol. 1, No. 1 (2012), pp. 53-74. - _N. J. A. Sloane_, Dec 25 2012.
Udita N. Katugampola, A new Fractional Derivative and its Mellin Transform, arXiv preprint arXiv:1106.0965 [math.CA], 2011.
Udita N. Katugampola, Mellin Transforms of the Generalized Fractional Integrals and Derivatives, arXiv preprint arXiv:1112.6031 [math.CA], 2011.
Donald E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78; arXiv:math/9207221 [math.CA], 1992.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Toufik Mansour and Matthias Schork, Generalized Bell numbers and algebraic differential equations, Pure Math. Appl.(PU. MA), Vol. 23, No. 2 (2012), pp. 131-142.
Toufik Mansour, Matthias Schork, and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.3.
Toufik Mansour and Mark Shattuck, A polynomial generalization of some associated sequences related to set partitions, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
Toufik Mansour, Augustine Munagi, and Mark Shattuck, Set partitions with colored singleton blocks, Discrete Mathematics Letters, 13. 100, (2024). See p. 100.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics, Vol. 13, No. 2 (2019), pp. 495-517.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Jose L. Ramirez and Mark Shattuck, A (p, q)-Analogue of the r-Whitney-Lah Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.6.
Mark Shattuck, Identities for Generalized Whitney and Stirling Numbers, Journal of Integer Sequences, Vol. 20 (2017), Article 17.10.4.
Mark Shattuck, Some formulas for the restricted r-Lah numbers, Annales Mathematicae et Informaticae, Vol. 49 (2018), Eszterházy Károly University Institute of Mathematics and Informatics, pp. 123-140.
Mark Shattuck, Combinatorial Proofs of Some Stirling Number Convolution Formulas, J. Int. Seq., Vol. 25 (2022), Article 22.2.2.
Michael Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq., Vol. 14 (2011) Article 11.9.7.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Same as A066667 and A105278 except for signs. Other variants: A111596 (differently signed triangle and (0,0)-based), A271703 (unsigned and (0,0)-based), A089231.
A293125 (row sums) and A000262 (row sums of unsigned triangle).
Columns 1-6 (unsigned): A000142, A001286, A001754, A001755, A001777, A001778.
A002868 gives maximal element (in magnitude) in each row.
A248045 (central terms, negated). A130561 is a natural refinement.
Cf. A007318, A048786, A001263, A008275, A008277, A048993, A094638, A130534.
Cf. A008296, A360177.

a008297 n k = a008297_tabl !! (n-1) !! (k-1)
a008297_row n = a008297_tabl !! (n-1)
a008297_tabl = [-1] : f [-1] 2 where
   f xs i = ys : f ys (i + 1) where
     ys = map negate $
          zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Sep 30 2014

A008297 := (n,m) -> (-1)^n*n!*binomial(n-1,m-1)/m!;

a[n_, m_] := (-1)^n*n!*Binomial[n-1, m-1]/m!; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012, after Maple *)
T[n_, n_] := (-1)^n; T[n_, k_]/;0Oliver Seipel, Dec 06 2024 *)

T(n, m) = (-1)^n*n!*binomial(n-1, m-1)/m!
for(n=1,9, for(m=1,n, print1(T(n,m)", "))) \\ Charles R Greathouse IV, Mar 09 2016

use bigint; use ntheory ":all"; my @L; for my $n (1..9) { push @L, map { stirling($n,$,3)*(-1)**$n } 1..$n; } say join(", ",@L); # _Dana Jacobsen, Mar 16 2017

def A008297_triangle(dim): # computes unsigned T(n, k).
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(2+2*k)*M[n-1,k]+((k+1)*(k+2))*M[n-1,k+1]
    return M
A008297_triangle(9) # Peter Luschny, Sep 19 2012

a(n, m) = (-1)^n*n!*A007318(n-1, m-1)/m!, n >= m >= 1.
a(n+1, m) = (n+m)*a(n, m)+a(n, m-1), a(n, 0) := 0; a(n, m) := 0, n < m; a(1, 1)=1.
a(n, m) = ((-1)^(n-m+1))*L(1, n-1, m-1) where L(1, n, m) is the triangle of coefficients of the generalized Laguerre polynomials n!*L(n, a=1, x). These polynomials appear in the radial l=0 eigen-functions for discrete energy levels of the H-atom.
|a(n, m)| = Sum_{k=m..n} |A008275(n, k)|*A008277(k, m), where A008275 = Stirling numbers of first kind, A008277 = Stirling numbers of second kind. - Wolfdieter Lang
If L_n(y) = Sum_{k=0..n} |a(n, k)|*y^k (a Lah polynomial) then the e.g.f. for L_n(y) is exp(x*y/(1-x)). - Vladeta Jovovic, Jan 06 2001
E.g.f. for the k-th column (unsigned): x^k/(1-x)^k/k!. - Vladeta Jovovic, Dec 03 2002
a(n, k) = (n-k+1)!*N(n, k) where N(n, k) is the Narayana triangle A001263. - Philippe Deléham, Jul 20 2003
From Shai Covo (green355(AT)netvision.net.il), Feb 02 2010: (Start)
We have the following expressions for the Lah polynomial L_n(y) = Sum_{k=0..n} |a(n, k)|*y^k -- exact generalizations of results in A000262 for A000262(n) = L_n(1):
1) L_n(y) = y*exp(-y)*n!*M(n+1,2,y), n >= 1, where M (=1F1) is the confluent hypergeometric function of the first kind;
2) L_n(y) = exp(-y)* Sum_{m>=0} y^m*[m]^n/m!, n>=0, where [m]^n = m*(m+1)*...*(m+n-1) is the rising factorial;
3) L_n(y) = (2n-2+y)L_{n-1}(y)-(n-1)(n-2)L_{n-2}(y), n>=2;
4) L_n(y) = y*(n-1)!*Sum_{k=1..n} (L_{n-k}(y) k!)/((n-k)! (k-1)!), n>=1. (End)
The row polynomials are given by D^n(exp(-x*t)) evaluated at x = 0, where D is the operator (1-x)^2*d/dx. Cf. A008277 and A035342. - Peter Bala, Nov 25 2011
n!C(-xD,n) = Lah(n,:xD:) where C(m,n) is the binomial coefficient, xD= x d/dx, (:xD:)^k = x^k D^k, and Lah(n,x) are the row polynomials of this entry. E.g., 2!C(-xD,2)= 2 xD + x^2 D^2. - Tom Copeland, Nov 03 2012
From Tom Copeland, Sep 25 2016: (Start)
The Stirling polynomials of the second kind A048993 (A008277), i.e., the Bell-Touchard-exponential polynomials B_n[x], are umbral compositional inverses of the Stirling polynomials of the first kind signed A008275 (A130534), i.e., the falling factorials, (x)_n = n! binomial(x,n); that is, umbrally B_n[(x).] = x^n = (B.[x])_n.
An operational definition of the Bell polynomials is (xD_x)^n = B_n[:xD:], where, by definition, (:xD_x:)^n = x^n D_x^n, so (B.[:xD_x:])_n = (xD_x)_n = :xD_x:^n = x^n (D_x)^n.
Let y = 1/x, then D_x = -y^2 D_y; xD_x = -yD_y; and P_n(:yD_y:) = (-yD_y)_n = (-1)^n (1/y)^n (y^2 D_y)^n, the row polynomials of this entry in operational form, e.g., P_3(:yD_y:) = (-yD_y)_3 = (-yD_y) (yD_y-1) (yD_y-2) = (-1)^3 (1/y)^3 (y^2 D_y)^3 = -( 6 :yD_y: +  6 :yD_y:^2 + :yD_y:^3 ) = - ( 6 y D_y + 6 y^2 (D_y)^2 + y^3 (D_y)^3).
Therefore, P_n(y) = e^(-y) P_n(:yD_y:) e^y = e^(-y) (-1/y)^n (y^2 D_y)^n e^y = e^(-1/x) x^n (D_x)^n e^(1/x) = P_n(1/x) and P_n(x) =  e^(-1/x) x^n (D_x)^n e^(1/x) = e^(-1/x) (:x D_x:)^n e^(1/x). (Cf. also A094638.) (End)
T(n,k) = Sum_{j=k..n} (-1)^j*A008296(n,j)*A360177(j,k). - Mélika Tebni, Feb 02 2023

A105278 Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 36, 12, 1, 120, 240, 120, 20, 1, 720, 1800, 1200, 300, 30, 1, 5040, 15120, 12600, 4200, 630, 42, 1, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1, 3628800, 16329600

Offset: 1

Miklos Kristof, Apr 25 2005

T(n,k) is the number of partially ordered sets (posets) on n elements that consist entirely of k chains. For example, T(4, 3)=12 since there are exactly 12 posets on {a,b,c,d} that consist entirely of 3 chains. Letting ab denote a<=b and using a slash "/" to separate chains, the 12 posets can be given by a/b/cd, a/b/dc, a/c/bd, a/c/db, a/d/bc, a/d/cb, b/c/ad, b/c/da, b/d/ac, b/d/ca, c/d/ab and c/d/ba, where the listing of the chains is arbitrary (e.g., a/b/cd = a/cd/b =...cd/b/a). - Dennis P. Walsh, Feb 22 2007
Also the matrix product |S1|.S2 of Stirling numbers of both kinds.
This Lah triangle is a lower triangular matrix of the Jabotinsky type. See the column e.g.f. and the D. E. Knuth reference given in A008297. - Wolfdieter Lang, Jun 29 2007
The infinitesimal matrix generator of this matrix is given in A132710. See A111596 for an interpretation in terms of circular binary words and generalized factorials. - Tom Copeland, Nov 22 2007
Three combinatorial interpretations: T(n,k) is (1) the number of ways to split [n] = {1,...,n} into a collection of k nonempty lists ("partitions into sets of lists"), (2) the number of ways to split [n] into an ordered collection of n+1-k nonempty sets that are noncrossing ("partitions into lists of noncrossing sets"), (3) the number of Dyck n-paths with n+1-k peaks labeled 1,2,...,n+1-k in some order. - David Callan, Jul 25 2008
Given matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = D where D(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 21 2008
An e.g.f. for the row polynomials of A(n,k) = T(n,k)*a(n-k) is exp[a(.)* D_x * x^2] exp(x*t) = exp(x*t) exp[(.)!*Lag(.,-x*t,1)*a(.)*x], umbrally, where [(.)! Lag(.,x,1)]^n = n! Lag(n,x,1) is a normalized Laguerre polynomial of order 1. - Tom Copeland, Aug 29 2008
Triangle of coefficients from the Bell polynomial of the second kind for f = 1/(1-x). B(n,k){x1,x2,x3,...} = B(n,k){1/(1-x)^2,...,(j-1)!/(1-x)^j,...} = T(n,k)/(1-x)^(n+k). - Vladimir Kruchinin, Mar 04 2011
The triangle, with the row and column offset taken as 0, is the generalized Riordan array (exp(x), x) with respect to the sequence n!*(n+1)! as defined by Wang and Wang (the generalized Riordan array (exp(x), x) with respect to the sequence n! is Pascal's triangle A007318, and with respect to the sequence n!^2 is A021009 unsigned). - Peter Bala, Aug 15 2013
For a relation to loop integrals in QCD, see p. 33 of Gopakumar and Gross and Blaizot and Nowak. - Tom Copeland, Jan 18 2016
Also the Bell transform of (n+1)!. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
Also the number of k-dimensional flats of the n-dimensional Shi arrangement. - Shuhei Tsujie, Apr 26 2019
The numbers T(n,k) appear as coefficients when expanding the rising factorials (x)^k = x(x+1)...(x+k-1) in the basis of falling factorials (x)k = x(x-1)...(x-k+1). Specifically, (x)^n = Sum{k=1..n} T(n,k) (x)k. - _Jeremy L. Martin, Apr 21 2021

			T(1,1) = C(1,1)*0!/0! = 1,
T(2,1) = C(2,1)*1!/0! = 2,
T(2,2) = C(2,2)*1!/1! = 1,
T(3,1) = C(3,1)*2!/0! = 6,
T(3,2) = C(3,2)*2!/1! = 6,
T(3,3) = C(3,3)*2!/2! = 1,
Sheffer a-sequence recurrence: T(6,2)= 1800 = (6/3)*120 + 6*240.
B(n,k) =
   1/(1-x)^2;
   2/(1-x)^3,  1/(1-x)^4;
   6/(1-x)^4,  6/(1-x)^5,  1/(1-x)^6;
  24/(1-x)^5, 36/(1-x)^6, 12/(1-x)^7, 1/(1-x)^8;
The triangle T(n,k) begins:
  n\k      1       2       3      4      5     6    7  8  9 ...
  1:       1
  2:       2       1
  3:       6       6       1
  4:      24      36      12      1
  5:     120     240     120     20      1
  6:     720    1800    1200    300     30     1
  7:    5040   15120   12600   4200    630    42    1
  8:   40320  141120  141120  58800  11760  1176   56  1
  9:  362880 1451520 1693440 846720 211680 28224 2016 72  1
  ...
Row n=10: [3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1]. - _Wolfdieter Lang_, Feb 01 2013
From _Peter Bala_, Feb 24 2025: (Start)
The array factorizes as an infinite product (read from right to left):
  /  1                \        /1             \^m /1           \^m /1           \^m
  |  2    1            |      | 0   1          |  |0  1         |  |1  1         |
  |  6    6   1        | = ...| 0   0   1      |  |0  1  1      |  |0  2  1      |
  | 24   36  12   1    |      | 0   0   1  1   |  |0  0  2  1   |  |0  0  3  1   |
  |120  240 120  20   1|      | 0   0   0  2  1|  |0  0  0  3  1|  |0  0  0  4  1|
  |...                 |      |...             |  |...          |  |...          |
where m = 2. Cf. A008277 (m = 1), A035342 (m = 3), A035469 (m = 4), A049029 (m = 5) A049385 (m = 6), A092082 (m = 7), A132056 (m = 8), A223511 - A223522 (m = 9 through 20), A001497 (m = -1), A004747 (m = -2), A000369 (m = -3), A011801 (m = -4), A013988 (m = -5). (End)

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Peter Bala, Factorising (r,b)-Stirling arrays
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, arXiv preprint arXiv:1105.3043 [math.CO], 2011, J. Int. Seq. 14 (2011) # 11.9.5
Jean-Paul Blaizot and Maciej A. Nowak, Large N_c confinement and turbulence, arXiv:0801.1859 [hep-th], 2008.
David Callan, Sets, Lists and Noncrossing Partitions, arXiv:0711.4841 [math.CO], 2007-2008.
Pietro Codara, Ottavio M. D'Antona, and Pavol Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
Tom Copeland, Mathemagical Forests, Addendum to Mathemagical Forests, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions, A Class of Differential Operators and the Stirling Numbers
Siad Daboul, Jan Mangaldan, Michael Z. Spivey and Peter Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., 86 (2013), 39-47.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 33.
G. H. E. Duchamp et al., Feynman graphs and related Hopf algebras, J. Phys. (Conf Ser) 30 (2006) 107-118.
Rajesh Gopakumar and David J. Gross, Mastering the master field, arXiv:hep-th/9411021, 1994.
Gábor Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4. - From _Tom Copeland_, Oct 01 2015
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Donald E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
MacTutor History of Mathematics archive: Biography of Ivo Lah.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019. See p. 18.
Michael Penn, Lah Numbers and an appearance of exponential generating functions, YouTube video, 2025.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019. See p. 4.
Tilman Piesk, Illustration of the first four rows
Kornelia Ufniarz and Grzegorz Siudem, Combinatorial origins of the canonical ensemble, arXiv:2008.00244 [math-ph], 2020. See p. 5.
Weiping Wang and Tianming Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Wikipedia, Lah number

Cf. A000262, A103194, A105220.
Triangle of Lah numbers (A008297) unsigned.
Cf. A111596 (signed triangle with extra n=0 row and m=0 column).
Cf. A130561 (for a natural refinement).
Cf. A094638 (for differential operator representation).
Cf. A248045 (central terms), A002868 (row maxima).
Cf, A059110.
Cf. A001263, A008277.
Cf. A089231 (triangle with mirrored rows).
Cf. A271703 (triangle with extra n=0 row and m=0 column).

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

a105278 n k = a105278_tabl !! (n-1) !! (k-1)
a105278_row n = a105278_tabl !! (n-1)
a105278_tabl = [1] : f [1] 2 where
   f xs i = ys : f ys (i + 1) where
     ys = zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Sep 30 2014, Mar 18 2013

/* As triangle */ [[Binomial(n,k)*Factorial(n-1)/Factorial(k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 31 2014

The triangle: for n from 1 to 13 do seq(binomial(n,k)*(n-1)!/(k-1)!,k=1..n) od;
the sequence: seq(seq(binomial(n,k)*(n-1)!/(k-1)!,k=1..n),n=1..13);
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> (n+1)!, 9); # Peter Luschny, Jan 27 2016

nn = 9; a = x/(1 - x); f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y a], {x, 0, nn}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 11 2011 *)
nn = 9; Flatten[Table[(j - k)! Binomial[j, k] Binomial[j - 1, k - 1], {j, nn}, {k, j}]] (* Jan Mangaldan, Mar 15 2013 *)
rows = 10;
t = Range[rows]!;
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
T[n_, n_] := 1; T[n_, k_] /;0Oliver Seipel, Dec 06 2024 *)

use ntheory ":all"; say join ", ", map { my $n=$; map { stirling($n,$,3) } 1..$n; } 1..9; # Dana Jacobsen, Mar 16 2017

T(n,k) = Sum_{m=n..k} |S1(n,m)|*S2(m,k), k>=n>=1, with Stirling triangles S2(n,m):=A048993 and S1(n,m):=A048994.
T(n,k) = C(n,k)*(n-1)!/(k-1)!.
Sum_{k=1..n} T(n,k) = A000262(n).
n*Sum_{k=1..n} T(n,k) = A103194(n) = Sum_{k=1..n} T(n,k)*k^2.
E.g.f. column k: (x^(k-1)/(1-x)^(k+1))/(k-1)!, k>=1.
Recurrence from Sheffer (here Jabotinsky) a-sequence [1,1,0,...] (see the W. Lang link under A006232): T(n,k)=(n/k)*T(n-1,m-1) + n*T(n-1,m). - Wolfdieter Lang, Jun 29 2007
The e.g.f. is, umbrally, exp[(.)!* L(.,-t,1)*x] = exp[t*x/(1-x)]/(1-x)^2 where L(n,t,1) = Sum_{k=0..n} T(n+1,k+1)*(-t)^k = Sum_{k=0..n} binomial(n+1,k+1)* (-t)^k / k! is the associated Laguerre polynomial of order 1. - Tom Copeland, Nov 17 2007
For this Lah triangle, the n-th row polynomial is given umbrally by
  n! C(B.(x)+1+n,n) = (-1)^n C(-B.(x)-2,n), where C(x,n)=x!/(n!(x-n)!),
  the binomial coefficient, and B_n(x)= exp(-x)(xd/dx)^n exp(x), the n-th Bell / Touchard / exponential polynomial (cf. A008277). E.g.,
  2! C(-B.(-x)-2,2) = (-B.(x)-2)(-B.(x)-3) = B_2(x) + 5*B_1(x) + 6 = 6 + 6x + x^2.
  n! C(B.(x)+1+n,n) = n! e^(-x) Sum_{j>=0} C(j+1+n,n)x^j/j! is a corresponding Dobinski relation. See the Copeland link for the relation to inverse Mellin transform. - Tom Copeland, Nov 21 2011
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A008277 (D = (1+x)*d/dx), A035342 (D = (1+x)^3*d/dx), A035469 (D = (1+x)^4*d/dx) and A049029 (D = (1+x)^5*d/dx). - Peter Bala, Nov 25 2011
T(n,k) = Sum_{i=k..n} A130534(n-1,i-1)*A008277(i,k). - Reinhard Zumkeller, Mar 18 2013
Let E(x) = Sum_{n >= 0} x^n/(n!*(n+1)!). Then a generating function is exp(t)*E(x*t) = 1 + (2 + x)*t + (6 + 6*x + x^2)*t^2/(2!*3!) + (24 + 36*x + 12*x^2 + x^3)*t^3/(3!*4!) + ... . - Peter Bala, Aug 15 2013
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of A059110; 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 23 2018
Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates the Narayana matrix A001263. - Tom Copeland, Sep 23 2020
T(n,k) = A089231(n,n-k). - Ron L.J. van den Burg, Dec 12 2021
T(n,k) = T(n-1,k-1) + (n+k-1)*T(n-1,k). - Bérénice Delcroix-Oger, Jun 25 2025

Stirling comments and e.g.f.s from Wolfdieter Lang, Apr 11 2007

A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).

Original entry on oeis.org

0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156, 56410454014314490981, 1399496554158060983080

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple grammar.
Number of {121,212}-avoiding n-ary words of length n. - Ralf Stephan, Apr 20 2004
The infinite continued fraction (1+n)/(1+(2+n)/(2+(3+n)/(3+...))) converges to the rational number A052852(n)/A000262(n) when n is a positive integer. - David Angell (angell(AT)maths.unsw.edu.au), Dec 18 2008
a(n) is the total number of components summed over all nilpotent partial permutations of [n]. - Geoffrey Critzer, Feb 19 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David Angell, A family of continued fractions, Journal of Number Theory, Volume 130, Issue 4, April 2010, Pages 904-911, Section 2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 820
Florent Hivert, Jean-Christophe Novelli and Jean-Yves Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
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 N-numbers, not their A-numbers.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.
Index entries for sequences related to Laguerre polynomials

Row sums of unsigned triangle A062139 (generalized a=2 Laguerre).
Cf. A000262. A000169, A000312.
Cf. A059114, A288268.

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

spec := [S,{B=Set(C),C=Sequence(Z,1 <= card),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a := n -> ifelse(n = 0, 0, n!*hypergeom([-n+1], [1], -1)): seq(simplify(a(n)), n = 0..18);  # Peter Luschny, Dec 30 2024

Table[n!*SeriesCoefficient[(x/(1-x))*E^(x/(1-x)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 09 2012 *)
Table[If[n==0, 0, n!*LaguerreL[n-1, 0, -1]], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *)

my(x='x+O('x^30)); concat([0], Vec(serlaplace((x/(1-x))*exp(x/(1-x))))) \\ G. C. Greubel, May 15 2018

[0 if n==0 else factorial(n)*gen_laguerre(n-1, 0, -1) for n in (0..30)] # G. C. Greubel, Feb 23 2021

D-finite with recurrence: a(1)=1, a(0)=0, (n^2+2*n)*a(n)+(-4-2*n)*a(n+1)+ a(n+2)=0.
a(n) = Sum_{m=0..n} n!*binomial(n+2, n-m)/m!. - Wolfdieter Lang, Jun 19 2001
a(n) = n*A002720(n-1). [Riordan] - Vladeta Jovovic, Mar 18 2005
Related to an n-dimensional series: for n>=1, a(n) = (n!/e)*Sum_{k_n>=k_{n-1}>=...>=k_1>=0} 1/(k_n)!. - Benoit Cloitre, Sep 30 2006
E.g.f.: (x/(1-x))*exp((x/(1-x))) = (x/(1-x))*G(0); G(k)=1+x/((2*k+1)*(1-x)-x*(1-x)*(2*k+1)/(x+(1-x)*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A000262 and A005493. - Peter Bala, Nov 25 2011
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+1/4)/sqrt(2). - Vaclav Kotesovec, Oct 09 2012
a(n) = n!*hypergeom([-n+1], [1], -1) for n>=1. - Peter Luschny, Oct 18 2014 [Simplified by Natalia L. Skirrow, 30 December 2024]
a(n) = Sum_{k=0..n} L(n,k)*k; L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014
a(n) = n!*LaguerreL(n-1, 0, -1) for n>=1. - Peter Luschny, Apr 08 2015, simplified Dec 30 2024
The series reversion of the e.g.f. equals W(x)/(1 + W(x)) = x - 2^2*x^2/2! + 3^3*x^3/3! - 4^4*x^4/4! + ..., essentially the e.g.f. for a signed version of A000312, where W(x) is Lambert's W-function (see A000169). - Peter Bala, Jun 14 2016
Equals column A059114(n, 2) for n >= 1. - G. C. Greubel, Feb 23 2021
a(n) = Sum_{k=1..n} k * A271703(n,k). - Geoffrey Critzer, Feb 19 2022

A048854 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].

Original entry on oeis.org

1, 2, 1, 12, 12, 1, 120, 180, 30, 1, 1680, 3360, 840, 56, 1, 30240, 75600, 25200, 2520, 90, 1, 665280, 1995840, 831600, 110880, 5940, 132, 1, 17297280, 60540480, 30270240, 5045040, 360360, 12012, 182, 1, 518918400, 2075673600, 1210809600, 242161920, 21621600, 960960, 21840, 240, 1, 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1

Offset: 0

Wolfdieter Lang

s(n,x) := Sum_{m=0..n} T(n,m)*x^m are monic polynomials satisfying 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} A048786(n,m)*x^m (row polynomials of triangle A048786) and p(0,x)=1.
In the umbral calculus (see the Roman reference, p. 21) the s(n,x) are called Sheffer polynomials for(1/sqrt(1+4*t),t/(1+4*t)). Here the Sheffer notation differs. See the W. Lang link under A006232.
For the general L[d,a] triangles see A286724, also for references.
This is the generalized signless Lah number triangle L[4,1], the Sheffer triangle ((1 - 4*t)^(-1/2), t/(1 - 4*t)). It is defined as transition matrix
risefac[4,1](x, n) = Sum_{m=0..n} L[4,1](n, m)*fallfac[4,1](x, m), where risefac[4,1](x, n) := Product_{0..n-1} (x  + (1 + 4*j)) for n >= 1 and risefac[4,1](x, 0) := 1, and  fallfac[4,1](x, n) := Product_{0..n-1} (x - (1 + 4*j)) for n >= 1 and fallfac[4,1](x, 0) := 1.
In matrix notation: L[4,1] = S1phat[4,1]*S2hat[4,1] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A290319 and A111578 (but here with offsets 0), respectively.
The a- and z-sequences for this Sheffer matrix have e.g.f.s Ea(t) = 1 + 4*t and Ez(t) = (1 + 4*t)*(1 - (1 + 4*t)^(-1/2))/t, respectively. That is, a = {1, 4, repeat(0)} and z(n) = 2*A292220(n). See the W. Lang link on a- and z-sequences there.
The inverse matrix T^(-1) = L^(-1)[4,1] is Sheffer ((1 + 4*t)^(-1/2), t/(1 + 4*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
fallfac[4,1](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[4,1](x, m), n >= 0.
Diagonal sequences have o.g.f. G(d, x) = A001813(d)*Sum_{m=0..d} A091042(d, m)*x^m/(1 - x)^{2*d + 1}, for d >= 0 (d=0 main diagonal). G(d, x) generates {A001813(d)*binomial(2*(n + d),2*d)}{n >= 0}. See the second W. Lang link on how to compute o.g.f.s of diagonal sequences of general Sheffer triangles. - _Wolfdieter Lang, Oct 12 2017

			The triangle T(n, m) begins:
n\m         0          1          2         3        4      5     6   7 8  ...
0:          1
1:          2          1
2:         12         12          1
3:        120        180         30         1
4:       1680       3360        840        56        1
5:      30240      75600      25200      2520       90      1
6:     665280    1995840     831600    110880     5940    132     1
7:   17297280   60540480   30270240   5045040   360360  12012   182   1
8:  518918400 2075673600 1210809600 242161920 21621600 960960 21840 240 1
...
n = 9: 17643225600 79394515200 52929676800 12350257920 1323241920 73513440 2227680 36720 306 1,
n = 10: 670442572800 3352212864000 2514159648000 670442572800 83805321600 5587021440 211629600 4651200 58140 380 1.
...
Recurrence from a-sequence: T(4, 2) = 2*T(3, 1) + 4*4*T(3, 2) = 2*180 + 16*30 = 840.
Recurrence from z-sequence: T(4, 0) = 4*(z(0)*T(3, 0) + z(1)*T(3, 1) + z(2)*T(3, 2)+ z(3)*T(3, 3)) = 4*(2*120 + 2*180 - 8*30 + 60*1) = 1680.
Four term recurrence: T(4, 2) = T(3, 1) + 2*13*T(3, 2) - 8*3*5*T(2, 2) =  180 + 26*30 - 120*1 = 840.
Meixner type identity for n = 2: (D_x - 4*(D_x)^2)*(12 + 12*x + 1*x^2) = (12 + 2*x) - 4*2 = 2*(2 + x).
Sheffer recurrence for R(3, x): [(2 + x) + 8*(1 + x)*D_x + 16*x*(D_x)^2] (12 + 12*x + 1*x^2) = (2 + x)*(12 + 12*x + x^2) + 8*(1 + x)*(12 + 2*x) + 16*2*x = 120 + 180*x + 30*x^2 + x^3 = R(3, x).
Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*10/2)*(1*30/3! + 4*1/2!) = 840.
Diagonal sequence d = 2: {12, 180, 840 ...} has o.g.f. 12*(1 + 10*x + 5*x^2)/(1 - x)^5  (see A001813(2) and row n=2 of A091042) generating
{12*binomial(2*(n + 2), 4)}_{n >= 0}. - _Wolfdieter Lang_, Oct 12 2017

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Orli Herscovici, Ross G. Pinsky, An identity involving stirling numbers of both kinds and its connection to right-to-left minima of certain set partitions, (2019).
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017, section C) 4.
Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Related to triangle A046521. Cf. A048786. a(n, 0) = A001813.
A111578, A271703 L[1,0], A286724 L[2,1], A290319, A290596 L[3,1], A290597 L[3,2], A292220.
The diagonal sequences are: A000012, 2*A000384(n+1), 12*A053134, 120*A053135, 1680*A053137, ... - Wolfdieter Lang, Oct 12 2017

A290604_row := proc(n) exp(x*t/(1-4*t))/sqrt(1-4*t): series(%, t, n+2): seq(n!*coeff(coeff(%,t,n),x,j), j=0..n) end: seq(A290604_row(n), n=0..9); # Peter Luschny, Sep 23 2017

T[n_, m_] := n!/m! * Binomial[2*n, n] * Binomial[n, m] / Binomial[2*m, m]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)
T[0, 0] = 1; T[-1, ] = T[, -1] = 0; T[n_, m_] /; n < m = 0; T[n_, m_] := T[n, m] = T[n-1, m-1] + 2*(4*n-3)*T[n-1, m] - 8*(n-1)*(2*n-3)*T[n-2, m]; Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Sep 23 2017 *)

T(n, m) = (n!/m!)*A046521(n, m) = (n!/m!)* binomial(2*n, n)*binomial(n, m)/binomial(2*m, m), n >= m >= 0, a(n, m) := 0, n < m.
Sum_{n>=0, k>=0} T(n, k)*x^n*y^k/(2*n)! = exp(x)*cosh(sqrt(x*y)). - Vladeta Jovovic, Feb 21 2003
T(n, m) = L[4,1](n,m) = Sum_{k=m..n} A290319(n, k)*A111578(k+1, m+1), 0 <= m <= n.
E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
(1 - 4*t)^(-1/2)*exp(x*t/(1 - 4*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 4*t)^(-1/2)*(t/(1 - 4*t))^m/m!, m >= 0.
Three term recurrence for column entries m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 4*n*T(n-1, m) with T(n, m) = 0 for n < m, and for the column m = 0: T(n, 0) = n*Sum_{j=0..n-1} z(j)*T(n-1, j), n >= 1, T(0, 0) = 0, from the a-sequence {1, 4 repeat(0)} and the z(j) = 2*A292220(j) (see above).
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(4*n - 3)*T(n-1, m) - 8*(n-1)*(2*n - 3)*T(n-2, m), n >= m >= 0, with T(0, 0) =1, T(-1, m) = 0, T(n, -1) = 0 and T(n, m) = 0 if n < m.
Meixner type identity for (monic) row polynomials: (D_x/(1 + 4*D_x)) * R(n, x) = n * R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx. That is, Sum_{k=0..n-1} (-4)^k*(D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1.
General recurrence for Sheffer row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
R(n, x) = [(2 + x)*1 + 8*(1 + x)*D_x + 16*x*(D_x)^2]*R(n-1, x), n >= 1, with R(0, x) = 1.
Boas-Buck recurrence for column m (see a comment in A286724 with references): T(n, m) = (n!/(n-m))*(2 + 4*m)*Sum_{p=0..n-1-m} 4^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.
Explicit form (from the o.g.f.s of diagonal sequences): ((2*(n-m))!/(n-m)!)*binomial(2*n,2*(n-m)), n >= m >= 0, and vanishing for n < m. - Wolfdieter Lang, Oct 12 2017

Name changed, after merging my newer duplicate, from Wolfdieter Lang, Oct 10 2017

A084357 Number of sets of sets of lists.

Original entry on oeis.org

1, 1, 4, 23, 171, 1552, 16583, 203443, 2813660, 43258011, 731183365, 13466814110, 268270250977, 5744515120489, 131525839441428, 3205279987587275, 82812074976214547, 2260364854328771548, 64979726427408468055, 1961976154991285214707, 62065551492895731512852

Offset: 0

N. J. A. Sloane, Jun 22 2003

nonn

In the book by Flajolet and Sedgewick on page 139 incorrectly gives a(5) = 1542. - Vaclav Kotesovec, Jul 11 2020

T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

T. D. Noe, Table of n, a(n) for n = 0..100
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 139.
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem [J. Phys. A 37 (2004), 3475-3487]
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Row sums of A079005 and row sums of A088814.

Cf. A000110, A008297, A271703.

with(combstruct); SetSetSeqL := [T, {T=Set(S), S=Set(U,card >= 1), U=Sequence(Z,card >=1)},labeled]; [seq(count(%,size=j),j=1..12)];

a[n_] = Sum[ n!/k!*Binomial[n-1, k-1]*BellB[k], {k, 0, n}]; a[0] = 1; Array[a, 20, 0]
(* Jean-François Alcover, Jun 22 2011, after Vladeta Jovovic *)

E.g.f.: exp(exp(x/(1-x))-1). Lah transform of Bell numbers: Sum_{k=0..n} n!/k!*binomial(n-1, k-1)*Bell(k). - Vladeta Jovovic, Sep 28 2003

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A286724 Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].

Original entry on oeis.org

1, 2, 1, 8, 8, 1, 48, 72, 18, 1, 384, 768, 288, 32, 1, 3840, 9600, 4800, 800, 50, 1, 46080, 138240, 86400, 19200, 1800, 72, 1, 645120, 2257920, 1693440, 470400, 58800, 3528, 98, 1, 10321920, 41287680, 36126720, 12042240, 1881600, 150528, 6272, 128, 1, 185794560, 836075520, 836075520, 325140480, 60963840, 6096384, 338688, 10368, 162, 1, 3715891200, 18579456000, 20901888000, 9289728000, 2032128000, 243855360, 16934400, 691200, 16200, 200, 1

Offset: 0

Wolfdieter Lang, Jun 16 2017

These generalized unsigned Lah numbers are the instance L[2,1] of the Sheffer triangles called L[d,a], with integers d >= 1 and integers 0 <= a < d with gcd(d,a) = 1. The standard unsigned Lah numbers are L[1,0] = A271703.
The Sheffer structure of L[d,a] is ((1 - d*t)^(-2*a/d), t/(1 - d*t)). This follows from the defining property
  risefac[d,a](x, n) = Sum_{m=0..n} L[d,a](n, m)*fallfac[d,a](x, m), where risefac[d,a](x, n):= Product_{0..n-1} (x  + (a+d*j)) for n >= 1 and risefac[d,a](x, 0) := 1, and fallfac[d,a](x, n):= Product_{0..n-1} (x - (a+d*j)) = for n >= 1 and fallfac[d,a](x, 0) := 1. Such rising and falling factorials arise in the generalization of Stirling numbers of both kinds S2[d,a] and S1[d,a]. See the Peter Bala link under A143395 for these falling factorials called there [t;a,b,c]_n with t=x, a=d, b=0, c=a.
In matrix notation: L[d,a] = S1phat[d,a].S2hat[d,a] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations with Sheffer structures S1phat[d,a] = ((1 - d*t)^(-a/d), -(1/d)*(log(1 - d*t))) and S2hat[d,a] = (exp(a*t), (1/d)*(exp(d*t) - 1). See, e.g., S1phat[2,1] = A028338 and S2hat[2,1] = A039755.
The a- and z-sequences for these Sheffer matrices have e.g.f.s 1 + d*t and ((1 + d*t)/t)*(1 - (1 + d*t)^(-2*a/d)), respectively. See a W. Lang link under A006232 for these types of sequences.
E.g.f. of row polynomials R[d,a](n, x) := Sum_{m=0..n} L[d,a](n, m)*x^m
  (1 - d*x)^(-2*a/d)*exp(t*x/(1 - d*x)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - d*t)^(-2*a/d)*(t/(1 - d*t))^m/m, m >= 0.
Meixner type identity for (monic) row polynomials: (D_x/(1 + d*D_x)) * R[d,a](n, x) = n*R[d,a](n-1, x), n >= 1, with R[d,a](0, x) = 1. The series in the differentiations D_x = d/dx terminates.
General Sheffer recurrence for row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
  R[d,a](n, x) = [(2*a+x)*1 + 2*d*(a + x)*D_x + d^2*x*(D_x)^2]*R[d,a](n-1, x), n >= 1, with R[d,a](0, x) = 1.
The inverse matrix L^(-1)[d,a] is Sheffer (g[d,a](-t), -f[d,a](-t)) with L[d,a] Sheffer (g[d,a](t), f[d,a](t)) from above. This means (see the column e.g.f. of Sheffer matrices) that L^(-1)[d,a](n, m) = (-1)^(n-m)*L[d,a](n, m). Therefore, the recurrence relations can easily be rewritten for L^(-1)[d,a] by replacing a -> -a and d -> -d.
fallfac[d,a](x, n) = Sum_{m=0..n} L^(-1)[d,a](n, m)*risefac[d,a](x, m), n >= 0.
From Wolfdieter Lang, Aug 12 2017: (Start)
The Sheffer row polynomials R[d,a](n, x) belong to the Boas-Buck class and satisfy therefore the Boas-Buck identity (see the reference, and we use the notation of Rainville, Theorem 50, p. 141, adapted to an exponential generating function) (E_x - n*1)*R[d,a](n, x) = - n*(2*a*1 + d*E_x) * Sum_{k=0..n-1} d^k*R(d,a;n-1-k,x)/(n-1-k)!, with E_x = x*d/dx (Euler operator).
This implies a recurrence for the sequence of column m: L[d,a](n, m) = (n!*(2*a + d*m)/(n-m))*Sum_{p=0..n-1-m} d^p*L[d,a](n-1-p, m)/(n-1-p)!, for n > m>=0, and input L[d,a](m, m) = 1. For the present [d,a] = [2,1] instance see the formula and example sections. (End)
From Wolfdieter Lang, Sep 14 2017: (Start)
The diagonal sequences are 2^D*D!*(binomial(m+D, m))^2, m >= 0, for D >= 0 (main diagonal D = 0). From the o.g.f.s obtained via Lagrange's theorem. See the second W. Lang link below for the general Sheffer case.
The o.g.f. of the diagonal D sequence is 2^D*D!*Sum_{m=0..D} A008459(D, m)*x^m /(1- x)^(2*D + 1),  D >= 0. (End)
It appears that this is also the matrix square of unsigned triangle of coefficients of Laguerre polynomials n!*L_n(x), abs(A021009(n, k)). - Ali Pourzand, Mar 10 2025 [This observation is correct. - Peter Luschny, Mar 10 2025]

			The triangle T(n, m) begins:
  n\m        0         1         2         3        4       5      6     7   8 9
  0:         1
  1:         2         1
  2:         8         8         1
  3:        48        72        18         1
  4:       384       768       288        32        1
  5:      3840      9600      4800       800       50       1
  6:     46080    138240     86400     19200     1800      72      1
  7:    645120   2257920   1693440    470400    58800    3528     98     1
  8:  10321920  41287680  36126720  12042240  1881600  150528   6272   128   1
  9: 185794560 836075520 836075520 325140480 60963840 6096384 338688 10368 162 1
  ...
From _Wolfdieter Lang_, Aug 12 2017: (Start)
Recurrence for column elements with m >= 1, and input column m = 0: T(3, 2) = (3/2)*T(2, 1) + 2*3*T(2, 2) = (3/2)*8 + 6 = 18.
Four term recurrence: T(3, 2) = T(2, 1) + 2*5*T(2, 2) - 4*2^2*T(1, 2) = 8 + 10 + 0 = 18.
Meixner type identity, n=2: 2*R(1, x) = (D_x - 2*(D_x)^2)*R(2, x), 2*(2 + x) = (8 + 2*x) - 2*2.
Sheffer recurrence: R(2, x) = (2 + x)*(2 + x) + 4*(1 + x)*1 + 0 = 8 + 8*x + x^2.
Boas-Buck recurrence for column m = 2 and n = 4: T(4, 2) = (2*4!*3/2)*(1*T(3, 2)/3! + 2*T(2, 2)/2!) = 4!*3*(18/3! + 1) = 288. (End)
Diagonal sequence D = 1: o.g.f. 2*1!*(1 + 1*x)/(1- x)^3 generating
{2*(binomial(m+1, m))^2}_{m >= 0} = {2, 8, 18, 32, ...}. - _Wolfdieter Lang_, Sep 14 2017

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).
Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.
Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.

Peter Bala, The white diamond product of power series
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Cf. A006232, A025167, A028338, A039755, A143395, A271703, A021009.
Column sequences (no leading zeros): A000165, A014479, A286725.
Diagonal sequences: A000012, 2*A000290(m+1), 8*A000537(n+1), 48*A001249, 384*A288876. - Wolfdieter Lang, Sep 14 2017
Row sums are A025167. - Michael Somos, Sep 27 2017

T := (n, k) -> ifelse(n < k, 0, ifelse(k = 0, n!*2^n, (n/k)*T(n-1, k-1) + 2*n*T(n-1, k))): seq(seq(T(n, k), k = 0..n), n = 0..10);  # Peter Luschny, Mar 10 2025

T[ n_, k_] := Coefficient[ Integrate[ Exp[-x^2 - y x] HermiteH[n, x]^2, {x, -Infinity, Infinity}] / (Sqrt[Pi] Exp[y^2 / 4]), y, 2 k]; (* Michael Somos, Sep 27 2017 *)

# Using the function A021009_triangle, displays as a matrix. Following the observation of Ali Pourzand.
print(A021009_triangle(9)^2)  # Peter Luschny, Mar 10 2025

T(n, m) = L[2,1](n, m) = Sum_{k=m..n} A028338(n, k)*A039755(k, m).
Three term recurrence for column elements with m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 2*n*T(n-1, m) with T(n, m) = 0 for n < m and the column m = 0 is T(n, 0) = (2*n)!! = n*2^n = A000165(n). (From the a- and z-sequences {1, 2, repeat(0)} and {2, repeat(0)}, respectively.)
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(2*n-1)*T(n-1, m) - 4*(n-1)^2*T(n-2, m), n >= m >= 0, with T(0, 0) = 1, T(-1, m) = 0, T(n, -1) = 0 and T(n, m) = 0 if n < m.
E.g.f. of row polynomials R(n, x) = R[2,1](n, x) (i.e., e.g.f. of the triangle): (1/(1-2*t))*exp(x*t/(1-2*t)).
E.g.f. of column m sequences: (t^m/(1-2*t)^(m+1))/m!, m >= 0.
Meixner type identity: Sum_{k=0..n-1} (-1)^k*2^k*(D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx.
Sheffer recurrence: R(n, x) = [(2 + x)*1 + 4*(1 + x)*D_x + 4*x*(D_x)^2]*R(n-1, x), n >= 1, and R(0, x) = 1.
Boas-Buck recurrence for column m (see a comment above): T(n, m) = (2*n!*(1 + m)/(n-1))*Sum_{p=0..n-1-m} 2^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, and input T(m, m) = 1. - Wolfdieter Lang, Aug 12 2017
Explicit form (from the diagonal sequences with the o.g.f.s given as a comment above): T(n, m) = 2^(n-m)*(n-m)!*(binomial(n, n-m))^2 for n >= m >= 0. - Wolfdieter Lang, Sep 23 2017
Let R(n,x) denote the n-th row polynomial. Then x^n*R(n,x) = x^n o x^n, where o denotes the deformed Hadamard product of power series defined in Bala, Section 3.1. - Peter Bala, Jan 18 2018

cp's OEIS Frontend

A344051 a(n) = Sum_{k=0..n} binomial(n, k)*|Lah(n, k)|. Binomial convolution of the unsigned Lah numbers A271703.

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A356654 Triangle read by rows. T(n, k) = k! * Sum_{j=k..n} Lah(n, j) * Stirling2(j, k), where Lah(n, k) = A271703(n, k).

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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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A008297 Triangle of Lah numbers.

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A105278 Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.

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