A001754 -id:A001754 - cp's OEIS frontend

A008297 Triangle of Lah numbers.

-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

A111596 The matrix inverse of the unsigned Lah numbers A271703.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 6, -6, 1, 0, -24, 36, -12, 1, 0, 120, -240, 120, -20, 1, 0, -720, 1800, -1200, 300, -30, 1, 0, 5040, -15120, 12600, -4200, 630, -42, 1, 0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1, 0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1

Offset: 0

Wolfdieter Lang, Aug 23 2005

Also the associated Sheffer triangle to Sheffer triangle A111595.
Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-1,x), which equals (-1)^n * Lag(n,x,-1) below. Lag(n,Lag(.,x,-1),-1) = x^n evaluated umbrally, i.e., with (Lag(.,x,-1))^k = Lag(k,x,-1). - Tom Copeland, Apr 26 2014
Without row n=0 and column m=0 this is, up to signs, the Lah triangle A008297.
The unsigned column sequences are (with leading zeros): A000142, A001286, A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10.
The row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m, together with the row polynomials s(n,x) of A111595 satisfy the exponential (or binomial) convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), n>=0.
Exponential Riordan array [1,x/(1+x)]. Inverse of the exponential Riordan array [1,x/(1-x)], which is the unsigned version of A111596. - Paul Barry, Apr 12 2007
For the unsigned subtriangle without column number m=0 and row number n=0, see A105278.
Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices.
The unsigned row polynomials are Lag(n,-x,-1), the associated Laguerre polynomials of order -1 with negated argument. See Gradshteyn and Ryzhik, Abramowitz and Stegun and Rota (Finite Operator Calculus) for extensive formulas. - Tom Copeland, Nov 17 2007, Sep 09 2008
An infinitesimal matrix generator for unsigned A111596 is given by A132792. - Tom Copeland, Nov 22 2007
From the formalism of A132792 and A133314 for n > k, unsigned A111596(n,k) = a(k) * a(k+1)...a(n-1) / (n-k)! = a generalized factorial, where a(n) = A002378(n) = n-th term of first subdiagonal of unsigned A111596. Hence Deutsch's remark in A002378 provides an interpretation of A111596(n,k) in terms of combinations of certain circular binary words. - Tom Copeland, Nov 22 2007
Given T(n,k)= A111596(n,k) and 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 = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 27 2008
Operationally, the unsigned row polynomials may be expressed as p_n(:xD:) = x*:Dx:^n*x^{-1}=x*D^nx^n*x^{-1}= n!*binomial(xD+n-1,n) = (-1)^n n! binomial(-xD,n) = n!L(n,-1,-:xD:), where, by definition, :AB:^n = A^nB^n for any two operators A and B, D = d/dx, and L(n,-1,x) is the Laguerre polynomial of order -1. A similarity transformation of the operators :Dx:^n generates the higher order Laguerre polynomials, which can also be expressed in terms of rising or falling factorials or Kummer's confluent hypergeometric functions (cf. the Mathoverflow post). -  Tom Copeland, Sep 21 2019

			Binomial convolution of row polynomials: p(3,x) = 6*x-6*x^2+x^3; p(2,x) = -2*x+x^2, p(1,x) = x, p(0,x) = 1,
together with those from A111595: s(3,x) = 9*x-6*x^2+x^3; s(2,x) = 1-2*x+x^2, s(1,x) = x, s(0,x) = 1; therefore
9*(x+y)-6*(x+y)^2+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = (6*y-6*y^2+y^3) + 3*x*(-2*y+y^2) + 3*(1-2*x+x^2)*y + 9*x-6*x^2+x^3.
From _Wolfdieter Lang_, Apr 28 2014: (Start)
The triangle a(n,m) begins:
n\m  0     1       2       3      4     5   6  7
0:   1
1:   0     1
2:   0    -2       1
3:   0     6      -6       1
4:   0   -24      36     -12      1
5:   0   120    -240     120    -20     1
6:   0  -720    1800   -1200    300   -30   1
7:   0  5040  -15120   12600  -4200   630 -42  1
...
For more rows see the link.
(End)

G. C. Greubel, Rows n=0..100 of triangle, flattened
Wolfdieter Lang, The first 11 rows of the triangle.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 4.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 6.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 20.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) 11.6.7.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers; Generators, Inversion, and Matrix, Binomial, and Integral Transforms; Lagrange a la Lah
A. Hennessy and P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Mathoverflow, Pochhammer symbol of a differential, and hypergeometric polynomials, a question posed by Emilio Pisanty and answered by Tom Copeland, 2012.
J. Taylor, Counting words with Laguerre polynomials, DMTCS Proc., Vol. AS, 2013, p. 1131-1142. [_Tom Copeland_, Jan 08 2016] [Broken link]
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96. [_Tom Copeland_, Dec 20 2018]
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Row sums: A111884. Unsigned row sums: A000262.
A002868 gives maximal element (in magnitude) in each row.
Cf. A130561 for a natural refinement.
Cf. A264428, A264429, A271703 (unsigned).
Cf. A008297, A089231, A105278 (variants).
Cf. A002378, A008277, A132792, A133314, A132792, A001263.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::odd, -(n+1)!, (n+1)!), 9); # Peter Luschny, Jan 27 2016

a[0, 0] = 1; a[n_, m_] := ((-1)^(n-m))*(n!/m!)*Binomial[n-1, m-1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)
T[ n_, k_] := (-1)^n n! Coefficient[ LaguerreL[ n, -1, x], x, k]; (* Michael Somos, Dec 15 2014 *)
rows = 9;
t = Table[(-1)^(n+1) n!, {n, 1, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}]  // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

{T(n, k) = if( n<1 || k<1, n==0 && k==0, (-1)^n * n! * polcoeff( sum(k=1, n, binomial( n-1, k-1) * (-x)^k / k!), k))}; /* Michael Somos, Dec 15 2014 */

lah_number = lambda n, k: factorial(n-k)*binomial(n,n-k)*binomial(n-1,n-k)
A111596_row = lambda n: [(-1)^(n-k)*lah_number(n, k) for k in (0..n)]
for n in range(10): print(A111596_row(n)) # Peter Luschny, Oct 05 2014

# uses[inverse_bell_transform from A264429]
def A111596_matrix(dim):
    fact = [factorial(n) for n in (1..dim)]
    return inverse_bell_transform(dim, fact)
A111596_matrix(10) # Peter Luschny, Dec 20 2015

E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0.
E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)).
a(n, m) = ((-1)^(n-m))*|A008297(n, m)| = ((-1)^(n-m))*(n!/m!)*binomial(n-1, m-1), n>=m>=1; a(0, 0)=1; else 0.
a(n, m) = -(n-1+m)*a(n-1, m) + a(n-1, m-1), n>=m>=0, a(n, -1):=0, a(0, 0)=1; a(n, m)=0 if n|a(n,m)| = Sum_{k=m..n} |S1(n,k)|*S2(k,m), n>=0. S2(n,m):=A048993. S1(n,m):=A048994. - Wolfdieter Lang, May 04 2007
From Tom Copeland, Nov 21 2011: (Start)
For this Lah triangle, the n-th row polynomial is given umbrally by
  (-1)^n n! binomial(-Bell.(-x),n), where Bell_n(-x)= exp(x)(xd/dx)^n exp(-x), the n-th Bell / Touchard / exponential polynomial with neg. arg., (cf. A008277). E.g., 2! binomial(-Bell.(-x),2) = -Bell.(-x)*(-Bell.(-x)-1) = Bell_2(-x)+Bell_1(-x) = -2x+x^2.
A Dobinski relation is (-1)^n n! binomial(-Bell.(-x),n)= (-1)^n n! e^x Sum_{j>=0} (-1)^j binomial(-j,n)x^j/j!= n! e^x Sum_{j>=0} (-1)^j binomial(j-1+n,n)x^j/j!. See the Copeland link for the relation to inverse Mellin transform. (End)
The n-th row polynomial is (-1/x)^n e^x (x^2*D_x)^n e^(-x). - Tom Copeland, Oct 29 2012
Let f(.,x)^n = f(n,x) = x!/(x-n)!, the falling factorial,and r(.,x)^n = r(n,x) = (x-1+n)!/(x-1)!, the rising factorial, then the Lah polynomials, Lah(n,t)= n!*Sum{k=1..n} binomial(n-1,k-1)(-t)^k/k! (extra sign factor on odd rows), give the transform Lah(n,-f(.,x))= r(n,x), and Lah(n,r(.,x))= (-1)^n * f(n,x). - Tom Copeland, Oct 04 2014
|T(n,k)| = Sum_{j=0..2*(n-k)} A254881(n-k,j)*k^j/(n-k)!. Note that A254883 is constructed analogously from A254882. - Peter Luschny, Feb 10 2015
The T(n,k) are the inverse Bell transform of [1!,2!,3!,...] and |T(n,k)| are the Bell transform of [1!,2!,3!,...]. See A264428 for the definition of the Bell transform and A264429 for the definition of the inverse Bell transform. - Peter Luschny, Dec 20 2015
Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates a shifted, signed Narayana matrix A001263. - Tom Copeland, Sep 23 2020

New name using a comment from Wolfdieter Lang by Peter Luschny, May 10 2021

A271703 Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 36, 12, 1, 0, 120, 240, 120, 20, 1, 0, 720, 1800, 1200, 300, 30, 1, 0, 5040, 15120, 12600, 4200, 630, 42, 1, 0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1

Offset: 0

Peter Luschny, Apr 14 2016

The Lah numbers can be seen as the case m=1 of the family of triangles T_{m}(n,k) = T_{m}(n-1,k-1)+(k^m+(n-1)^m)*T_{m}(n-1,k) (see the link 'Partition transform').

This is the Sheffer triangle (lower triangular infinite matrix) (1, x/(1-x)), an element of the Jabotinsky subgroup of the Sheffer group. - Wolfdieter Lang, Jun 12 2017

			As a rectangular array (diagonals of the triangle):
  1,      1,       1,       1,       1,       1,       ... A000012
  0,      2,       6,       12,      20,      30,      ... A002378
  0,      6,       36,      120,     300,     630,     ... A083374
  0,      24,      240,     1200,    4200,    11760,   ... A253285
  0,      120,     1800,    12600,   58800,   211680,  ...
  0,      720,     15120,   141120,  846720,  3810240, ...
A000007, A000142, A001286, A001754, A001755,  A001777.
The triangle T(n,k) begins:
n\k 0       1        2        3        4       5      6     7    8  9 10 ...
0:  1
1:  0       1
2:  0       2        1
3:  0       6        6        1
4:  0      24       36       12        1
5:  0     120      240      120       20       1
6:  0     720     1800     1200      300      30      1
7:  0    5040    15120    12600     4200     630     42     1
8:  0   40320   141120   141120    58800   11760   1176    56    1
9:  0  362880  1451520  1693440   846720  211680  28224  2016   72  1
10: 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90  1
...  - _Wolfdieter Lang_, Jun 12 2017

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., pp. 312, 552.
I. Lah, Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, Mitt.-Bl. Math. Statistik, 7:203-213, 1955.
T. Mansour, M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press, 2016

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
M. F. Hasler and Peter Luschny, Formulas for A271703, OEIS Wiki, Aug. 2017.
S. A. Joni, G.-C. Rota, and B. Sagan, From sets to functions: Three elementary examples, Discrete Mathematics, Volume 37, Issues 2-3, 1981, 193-202.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.
Peter Luschny, Lah numbers
Peter Luschny, Partition transform
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See. p. 18.
Piotr Miska and Maciej Ulas, On some properties of the number of permutations being products of pairwise disjoint d-cycles, arXiv:1904.03395 [math.NT], 2019.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Variants: A008297 the main entry for these numbers, A105278, A111596 (signed).
A000262 (row sums). Largest number of the n-th row in A002868.
Cf. A097805, A354795, A360177.

T := (n, k) -> `if`(n=k, 1, binomial(n-1,k-1)*n!/k!):
seq(seq(T(n, k), k=0..n), n=0..9);

T[n_, k_] := Binomial[n, k]*FactorialPower[n-1, n-k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

@cached_function
def T(n,k):
    if k<0 : return 0
    if k==n: return 1
    return T(n-1,k-1) + (k+n-1)*T(n-1,k)
for n in (0..8): print([T(n,k) for k in (0..n)])

For a collection of formulas see the 'Lah numbers' link.
T(n, k) = A097805(n, k)*n!/k! = (-1)^k*P_{n, k}(1,1,1,...) where P_{n, k}(s) is the partition transform of s.
T(n, k) = coeff(n! * P(n), x, k) with P(n) = (1/n)*(Sum_{k=0..n-1}(x(n-k)*P(k))), for n >= 1 and P(n=0) = 1, with x(n) = n*x. See A036039. - Johannes W. Meijer, Jul 08 2016
From Wolfdieter Lang, Jun 12 2017: (Start)
E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k (that is egf of the triangle) is exp(x*t/(1-t)) (a Sheffer triangle of the Jabotinsky type).
E.g.f. column k: (t/(1-t))^k/k!.
Three term recurrence: T(n, k) = T(n-1, k-1) + (n-1+k)*T(n, k-1), n >= 1, k = 0..n, with T(0, 0) =1, T(n, -1) = 0, T(n, k) = 0 if n < k.
T(n, k) = binomial(n, k)*fallfac(x=n-1, n-k), with fallfac(x, n) = Product_{j=0..(n-1)} (x - j), for n >= 1, and 0 for n = 0.
risefac(x, n) = Sum_{k=0..n} T(n, k)*fallfac(k), with risefac(x, n) = Product_{j=0..(n-1)} (x + j), for n >= 1, and 0 for n = 0.
See Graham et al., exercise 31, p. 312, solution p. 552. (End)
Formally, let f_n(x) = Sum_{k>n} (k-1)!*x^k; then f_n(x) = Sum_{k=0..n} T(n, k)* x^(n+k)*f_0^((k))(x), where ^((k)) stands for the k-th derivative. - Luc Rousseau, Dec 27 2020
T(n, k) = Sum_{j=k..n} A354795(n, j)*A360177(j, k). - Mélika Tebni, Feb 02 2023
T(n, k) = binomial(n, k)*(n-1)!/(k-1)! for n, k > 0. - Chai Wah Wu, Nov 30 2023

A053495 Triangle formed by coefficients of numerator polynomials defined by iterating f(u,v) = 1/u - x*v applied to a list of elements {1,2,3,4,...}.

Original entry on oeis.org

1, 1, -1, -1, 2, -2, 1, -4, 6, -6, -1, 6, -18, 24, -24, 1, -9, 36, -96, 120, -120, -1, 12, -72, 240, -600, 720, -720, 1, -16, 120, -600, 1800, -4320, 5040, -5040, -1, 20, -200, 1200, -5400, 15120, -35280, 40320, -40320, 1, -25, 300, -2400, 12600

Offset: 0

Wouter Meeussen, Jan 27 2001

			1, 1 - x, -1 + 2*x - 2*x^2, 1 - 4*x + 6*x^2 - 6*x^3, ...

Diagonals give A000142, A001563, A001286, A001809, A001754, A001810, A001755, A001811, A001777. Except for first term, row sums give negative of A058307.

Row sums of positive entries give A001053, those of negative entries give -1*A001040.

CoefficientList[ #, x ]&/@Numerator[ FoldList[ (1/#1-x#2)&, 1, Range[ 12 ] ]//Together ]
FoldList[(1/#1-x#2)&, 1, Range[4] ]//Together (a simpler version, which shows the rational functions)

Table[ (-1)^(r+c+1) binomial[Floor[(r+c)/2], Floor[(r-c)/2]] Floor[(r+c+1)/2]! / Floor[(r-c+1)/2]!, {r, 0, 7}, {c, 0, r}]

a[0] := -1; a[1] := 1-x; a[n_] := a[n]= n x a[n-1] + a[n-2] (matches sequence except for a[0]).

A263776 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 8, 2, 42, 45, 25, 7, 1, 132, 220, 198, 112, 44, 12, 2, 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1, 1430, 4368, 7280, 8400, 7460, 5392, 3262, 1664, 716, 256, 74, 16, 2, 4862, 18564, 38556, 56100, 63648, 59670, 47802, 33338, 20466, 11115

Offset: 0

Christian Stump, Oct 26 2015

Row sums give A000142.
First column gives A000108.
Also the number of permutations of [n] with k crossings (see Corteel, Proposition 4).
Also the number of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2). - Alois P. Heinz, Nov 14 2015

			Triangle begins:
0 :   1;
1 :   1;
2 :   2;
3 :   5,    1;
4 :  14,    8,    2;
5 :  42,   45,   25,    7,   1;
6 : 132,  220,  198,  112,  44,  12,   2;
7 : 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1;
...

Alois P. Heinz, Rows n = 0..50, flattened
A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001.
S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163.
FindStat - Combinatorial Statistic Finder, The number of nestings of a permutation, The number of crossings of a permutation
R. Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
Lucas Sá and Antonio M. García-García, The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos, arXiv:2104.07647 [hep-th], 2021.

Columns k=0-10 give: A000108, A002696, A094218, A094219, A120812, A120813, A120814, A120815, A120816, A264496, A264497.

Cf. A000142, A001754, A002620, A260665, A260670, A287328, A291722.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
       add(expand(b(u+j-1, o-j)*x^(j-1)), j=1..o))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 14 2015

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Expand[b[u+j-1, o-j]*x^(j-1)], {j, 1, o}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[ T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

Sum_{k>0} k * T(n,k) = A001754(n).

T(n,n) = A287328(n). - Alois P. Heinz, Aug 31 2017

More terms from Alois P. Heinz, Oct 26 2015

A260665 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 12-3; triangle T(n,k), n>=0, 0<=k<=(n-1)*(n-2)/2-[n=0], read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 15, 7, 1, 1, 52, 39, 13, 12, 2, 1, 1, 203, 211, 112, 103, 41, 24, 17, 5, 2, 1, 1, 877, 1168, 843, 811, 492, 337, 238, 122, 68, 39, 28, 8, 5, 2, 1, 1, 4140, 6728, 6089, 6273, 4851, 3798, 2956, 1960, 1303, 859, 594, 314, 204, 110, 64, 43, 17, 8, 5, 2, 1, 1

Offset: 0

Alois P. Heinz, Nov 14 2015

Patterns 1-23, 3-21, 32-1 give the same triangle.

			T(4,1) = 7: 1324, 1342, 2134, 2314, 2341, 3124, 4123.
T(4,2) = 1: 1243.
T(4,3) = 1: 1234.
T(5,3) = 12: 12534, 12543, 13245, 13425, 13452, 21345, 23145, 23415, 23451, 31245, 41235, 51234.
T(5,4) = 2: 12435, 12453.
T(5,5) = 1: 12354.
T(5,6) = 1: 12345.
Triangle T(n,k) begins:
0 :   1;
1 :   1;
2 :   2;
3 :   5,    1;
4 :  15,    7,   1,   1;
5 :  52,   39,  13,  12,   2,   1,   1;
6 : 203,  211, 112, 103,  41,  24,  17,   5,  2,  1,  1;
7 : 877, 1168, 843, 811, 492, 337, 238, 122, 68, 39, 28, 8, 5, 2, 1, 1;

Alois P. Heinz, Rows n = 0..50, flattened
A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001

Columns k=0-10 give: A000110, A092923, A264451, A264452, A264453, A264454, A264455, A264456, A264457, A264458, A264459.
Row sums give A000142.
Cf. A000217, A001754, A260670, A263776.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
       add(expand(b(u+j-1, o-j)*x^(o-j)), j=1..o))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);

b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1], {j, 1, u}] + Sum[Expand[b[u + j - 1, o - j]*x^(o - j)], {j, 1, o}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0] ]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)

Sum_{k>0} k * T(n,k) = A001754(n).

A260670 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 23-1; triangle T(n,k), n>=0, 0<=k<=A125811(n)-1, read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 15, 6, 3, 52, 32, 23, 10, 3, 203, 171, 152, 98, 62, 22, 11, 1, 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4, 4140, 5444, 6460, 6082, 5513, 4302, 3328, 2141, 1393, 780, 432, 187, 88, 24, 6, 21147, 32919, 43626, 46508, 46880, 41979, 36774

Offset: 0

Alois P. Heinz, Nov 14 2015

Patterns 1-32, 3-12, 21-3 give the same sequence.

			T(3,1) = 1: 231.
T(4,1) = 6: 1342, 2314, 2413, 2431, 3241, 4231.
T(4,2) = 3: 2341, 3412, 3421.
T(5,2) = 23: 13452, 14523, 14532, 23415, 23514, 23541, 24351, 25341, 32451, 34125, 34152, 34215, 35124, 35142, 35214, 35412, 35421, 42351, 43512, 43521, 52341, 53412, 53421.
T(5,3) = 10: 23451, 24513, 24531, 34251, 35241, 45123, 45132, 45213, 45312, 45321.
T(5,4) = 3: 34512, 34521, 45231.
Triangle T(n,k) begins:
0 :   1;
1 :   1;
2 :   2;
3 :   5,   1;
4 :  15,   6,   3;
5 :  52,  32,  23,  10,   3;
6 : 203, 171, 152,  98,  62,  22,  11,   1;
7 : 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4;

Alois P. Heinz, Rows n = 0..50, flattened
A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001

Columns k=0-10 give: A000110, A264460, A264461, A264462, A264463, A264464, A264465, A264466, A264467, A264468, A264469.
Row sums give A000142.
Cf. A001754, A125811, A260665, A263776.

b:= proc(u, o) option remember;
     `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
       add(expand(b(u+j-1, o-j)*x^u), j=1..o))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n,0)):
seq(T(n), n=0..10);

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Expand[b[u+j-1, o-j]*x^u], {j, 1, o}]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 16 2017, after Alois P. Heinz *)

Sum_{k>0} k * T(n,k) = A001754(n).

A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 4, 30, 60, 35, 0, 5, 60, 210, 280, 126, 0, 6, 105, 560, 1260, 1260, 462, 0, 7, 168, 1260, 4200, 6930, 5544, 1716, 0, 8, 252, 2520, 11550, 27720, 36036, 24024, 6435, 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310

Offset: 0

Peter Luschny, Mar 20 2015

The rows give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1) / (n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022

This is related to the cluster fans of type B (see Fomin and Zelevinsky reference) - F. Chapoton, Nov 17 2022.

			[1]
[0, 1]
[0, 2,   3]
[0, 3,  12,   10]
[0, 4,  30,   60,   35]
[0, 5,  60,  210,  280,  126]
[0, 6, 105,  560, 1260, 1260,  462]
[0, 7, 168, 1260, 4200, 6930, 5544, 1716]
.
R_0(x) = 1/(x-1)^0.
R_1(x) = 0/(x-1)^1 + 1/(x-1)^2.
R_2(x) = 0/(x-1)^2 + 2/(x-1)^3 + 3/(x-1)^4.
R_3(x) = 0/(x-1)^3 + 3/(x-1)^4 + 12/(x-1)^5 + 10/(x-1)^6.
Then k!*[x^k] R_n(x) is A001286(k+2) and A001754(k+3) for n = 2, 3 respectively.
.
Seen as an array A(n, k) = binomial(n + k, k)*binomial(n + 2*k - 1, n + k):
[0] 1, 1,   3,   10,    35,    126,     462, ...
[1] 0, 2,  12,   60,   280,   1260,    5544, ...
[2] 0, 3,  30,  210,  1260,   6930,   36036, ...
[3] 0, 4,  60,  560,  4200,  27720,  168168, ...
[4] 0, 5, 105, 1260, 11550,  90090,  630630, ...
[5] 0, 6, 168, 2520, 27720, 252252, 2018016, ...
[6] 0, 7, 252, 4620, 60060, 630630, 5717712, ...

Seiichi Manyama, Rows n = 0..139, flattened
F. Chapoton Enumerative Properties of Generalized Associahedra, Sém. Loth. Comb. B51b (2004).
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
S. Fomin and A. Zelevinsky Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3.
Yunlong Shen and Lixin Shen, Orthogonal Fourier-Mellin moments for invariant pattern recognition, J. Opt. Soc. Am. A 11 (6) (1994) p 1748-1757, eq. (6).

T(n, n) = C(2*n-1, n) = A001700(n-1).
T(n, n-1) = A005430(n-1) for n >= 1.
T(n, n-2) = A051133(n-2) for n >= 2.
T(n, 2) = A027480(n-1) for n >= 2.
T(2*n, n) = A208881(n) for n >= 0.
A002002 (row sums).
Cf. A000142, A001286, A001754, A001755, A001777, A182626, A266732 (row k=3), A008316, A033282, A063007, A345013, A269944.

T_row := proc(n) local egf, k, F, t;
if n=0 then RETURN(1) fi;
egf := (x/(1-x))^n/n!; t := diff(egf,[x$n]);
F := convert(t,parfrac,x);
# print(seq(k!*coeff(series(F,x,20),x,k),k=0..7));
# gives A000142, A001286, A001754, A001755, A001777, ...
seq(coeff(F,(x-1)^(-k)),k=n..2*n) end:
seq(print(T_row(n)),n=0..7);
# 2nd version by R. J. Mathar, Dec 18 2016:
A253283 := proc(n,k)
    binomial(n,k)*binomial(n+k-1,k-1) ;
end proc:

Table[Binomial[n, k] Binomial[n + k - 1, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 22 2017 *)

T(n,k) = binomial(n,k)*binomial(n+k-1,k-1);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Apr 29 2018

The exponential generating functions for the rows of the square array L(n,k) = ((n+k)!/n!)*C(n+k-1,n-1) (associated to the unsigned Lah numbers) are given by R_n(x) = Sum_{k=0..n} T(n,k)/(x-1)^(n+k).
T(n,k) = C(n,k)*C(n+k-1,k-1).
Sum_{k=0..n} T(n,k) = (-1)^n*hypergeom([-n,n],[1],2) = (-1)^n*A182626(n).
Row generating function: Sum_{k>=1} T(n,k)*z^k = z*n* 2F1(1-n,n+1 ; 2; -z). - R. J. Mathar, Dec 18 2016
From Peter Bala, Feb 22 2017: (Start)
G.f.: (1/2)*( 1 + (1 - t)/sqrt(1 - 2*(2*x + 1)*t + t^2) ) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n,x) = (1/2)*(LegendreP(n, 2*x + 1) - LegendreP(n-1, 2*x + 1)) for n >= 1.
The row polynomials are the black diamond product of the polynomials x^n and x^(n+1) (see Dukes and White 2016 for the definition of this product).
exp(Sum_{n >= 1} R(n,x)*t^n/n) = 1 + x*t + x*(1 + 2*x)*t^2 + x*(1 + 5*x + 5*x^2)*t^3 + ... is a g.f. for A033282, but with a different offset.
The polynomials P(n,x) := (-1)^n/n!*x^(2*n)*(d/dx)^n(1 + 1/x)^n begin 1, 3 + 2*x , 10 + 12*x + 3*x^2, ... and are the row polynomials for the row reverse of this triangle. (End)
Let Q(n, x) = Sum_{j=0..n} (-1)^(n - j)*A269944(n, j)*x^(2*j - 1) and P(x, y) = (LegendreP(x, 2*y + 1) - LegendreP(x-1, 2*y + 1)) / 2 (see Peter Bala above). Then n!*(n - 1)!*[y^n] P(x, y) = Q(n, x) for n >= 1. - Peter Luschny, Oct 31 2022
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 0} binomial(2*n-1, n)*(x*t)^n/(1 - t)^(2*n) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n, x) = [t^n] ( (1 - t)/(1 - (1 + x)*t) )^n.
It follows that for integer x, the sequence {R(n, x) : n >= 0} satisfies the Gauss congruences: R(n*p^r, x) == R(n*p^(r-1), x) (mod p^r) for all primes p and positive integers n and r.
R(n, -2) = (-1)^n * A002003(n) for n >= 1.
R(n, 3) = A299507(n). (End)

A156992 Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.

Original entry on oeis.org

1, 2, 2, 6, 12, 6, 24, 72, 72, 24, 120, 480, 720, 480, 120, 720, 3600, 7200, 7200, 3600, 720, 5040, 30240, 75600, 100800, 75600, 30240, 5040, 40320, 282240, 846720, 1411200, 1411200, 846720, 282240, 40320, 362880, 2903040, 10160640, 20321280, 25401600, 20321280, 10160640, 2903040, 362880

Offset: 1

Roger L. Bagula, Feb 20 2009

Partition {1,2,...,n} into m subsets, arrange (linearly order) the elements within each subset, then arrange the subsets. - Geoffrey Critzer, Mar 05 2010
From Dennis P. Walsh, Nov 26 2011: (Start)
Number of ways to arrange n different books in a k-shelf bookcase leaving no shelf empty.
There are n! ways to arrange the books in one long line. With ni denoting the number of books for shelf i, we have n = n1 + n2 + ... + nk. Since the number of compositions of n with k summands is binomial(n-1,k-1), we obtain T(n,k) = n!*binomial(n-1,k-1) for the number of ways to arrange the n books on the k shelves.
Equivalently, T(n,k) is the number of ways to stack n different alphabet blocks into k labeled stacks.
Also, T(n,k) is the number of injective functions f:[n]->[n+k] such that (i) the pre-image of (n+j) exists for j=1..k and (ii) f has no fixed points, that is, for all x, f(x) does not equal x.
T(n,k) is the number of labeled, rooted forests that have (i) exactly k roots, (ii) each root labeled larger than any nonroot, (iii) each root with exactly one child node, (iv) n non-root nodes, and (v) at most one child node for each node in the forest.
(End)
Essentially, the triangle given by (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 29 2011
T(n,j+k) = Sum_{i=j..n-k} binomial(n,i)*T(i,j)*T(n-i,k). - Dennis P. Walsh, Nov 29 2011

			The triangle starts:
      1;
      2,      2;
      6,     12,      6;
     24,     72,     72,      24;
    120,    480,    720,     480,     120;
    720,   3600,   7200,    7200,    3600,    720;
   5040,  30240,  75600,  100800,   75600,  30240,   5040;
  40320, 282240, 846720, 1411200, 1411200, 846720, 282240, 40320;
From _Dennis P. Walsh_, Nov 26 2011: (Start)
T(3,2) = 12 since there are 12 ways to arrange books b1, b2, and b3 on shelves <shelf1><shelf2>:
   <b1><b2,b3>, <b1><b3,b2>, <b2><b1,b3>, <b2><b3,b1>,
   <b3><b1,b2>, <b3><b2,b1>, <b2,b3><b1>, <b3,b2><b1>,
   <b1,b3><b2>, <b3,b1><b2>, <b1,b2><b3>, <b2,b1><b3>.
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 98

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers

Cf. A002866 (row sums).
Column 1 = A000142. Column 2 = A001286 * 2! = A062119. Column 3 = A001754 * 3!. Column 4 = A001755 * 4!. Column 5 = A001777 * 5!. Column 6 = A001778 * 6!. Column 7 = A111597 * 7!. Column 8 = A111598 * 8!. Cf. A105278. - Geoffrey Critzer, Mar 05 2010
T(2n,n) gives A123072.

[Factorial(n)*Binomial(n-1,k-1): k in [1..n], n in [1..10]]; // G. C. Greubel, May 10 2021

seq(seq(n!*binomial(n-1,k-1),k=1..n),n=1..10); # Dennis P. Walsh, Nov 26 2011
with(PolynomialTools): p := (n,x) -> (n+1)!*hypergeom([-n],[],-x);
seq(CoefficientList(simplify(p(n,x)),x),n=0..5); # Peter Luschny, Apr 08 2015

Table[n!*Binomial[n-1, k-1], {n,10}, {k,n}]//Flatten

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

E.g.f. for column k is (x/(1-x))^k. - Geoffrey Critzer, Mar 05 2010
T(n,k) = A000142(n)*A007318(n-1,k-1). - Dennis P. Walsh, Nov 26 2011
Coefficient triangle of the polynomials p(n,x) = (n+1)!*hypergeom([-n],[],-x). - Peter Luschny, Apr 08 2015

A055303 Number of labeled rooted trees with n nodes and 2 leaves.

Original entry on oeis.org

3, 36, 360, 3600, 37800, 423360, 5080320, 65318400, 898128000, 13172544000, 205491686400, 3399953356800, 59499183744000, 1098446469120000, 21341245685760000, 435361411989504000, 9305850181275648000, 208013121699102720000, 4853639506312396800000

Offset: 3

Christian G. Bower, May 11 2000

a(n+1) is the sum of the zero moments over all permutations of n. E.g., a(4) is [1,2,3].[0,1,2] + [1,3,2].[0,1,2] + [2,1,3].[0,1,2] + [2,3,1].[0,1,2] + [3,1,2].[0,1,2] + [3,2,1].[0,1,2] = 8 + 7 + 7 + 5 + 5 + 4 = 36. - Jon Perry, Feb 20 2004

a(n) is the number of pairs of elements (p(i),p(j)) in an n-permutation such that i > j and p(i) < p(j) where j is not equal to i-1. Loosely speaking, we could say: the number of inversions that are not descents. A055303 + A001286 = A001809. For example, a(3)=3 from the permutations (given in one line notation): (2,3,1), (3,1,2), (3,2,1) we have the pairs (1,2), (2,3) and (1,3) respectively. - Geoffrey Critzer, Jan 06 2013

Index entries for sequences related to rooted trees

Column 2 of A055302.

Cf. A000217, A001754.

seq(n!*(n-2)*(n-1)/4, n = 3..21); # Zerinvary Lajos, Apr 25 2008 [corrected by Georg Fischer, Aug 17 2021]
f:= gfun:-rectoproc({(n-3)*a(n) - (n^2-n)*a(n-1), a(3)=3}, a(n), remember): map(f, [$3..20]); # Georg Fischer, Aug 17 2021

With[{nn=20}, Drop[CoefficientList[Series[x^3/(2(1-x)^3), {x,0,nn}], x] * Range[0,nn]!, 3]] (* Harvey P. Dale, Nov 22 2012 *)

E.g.f.: x^3/(2*(1-x)^3).
a(n) = (n-2)!*t(n-2)*t(n-1) = (n-2)!*(n-2)*(n-1)^2*n/4 = n!*(n-2)*(n-1)/4 = n!*t(n-2)/2 where t = A000217. - Jon Perry, Feb 22 2004
D-finite with recurrence: (n-3)*a(n) - (n^2 - n)*a(n-1) = 0. - Georg Fischer, Aug 17 2021
a(n) = 3 * A001754(n). - Alois P. Heinz, Aug 17 2021

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A008297 Triangle of Lah numbers.

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A111596 The matrix inverse of the unsigned Lah numbers A271703.

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A271703 Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.

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A053495 Triangle formed by coefficients of numerator polynomials defined by iterating f(u,v) = 1/u - x*v applied to a list of elements {1,2,3,4,...}.

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A263776 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings.

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A260665 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 12-3; triangle T(n,k), n>=0, 0<=k<=(n-1)*(n-2)/2-[n=0], read by rows.

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