A066667 -id:A066667 - cp's OEIS frontend

A089231 Triangular array A066667 or A008297 unsigned and transposed.

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

Offset: 1

Philippe Deléham, Dec 10 2003

Row sums: A000262.
T(n, k) is also the number of nilpotent partial one-one bijections (of an n-element set) of height k (height(alpha) = |Im(alpha)|). - Abdullahi Umar, Sep 14 2008
T(n, k) is also the number of acyclic directed graphs on n labeled nodes with k-1 edges with all indegrees and outdegrees at most 1. - Felix A. Pahl, Dec 25 2012
For n > 1, the n-th derivative of exp(1/x) is of the form (exp(1/x)/x^(2*n))*(P(n-1,x)) where P(n-1,x) is a polynomial of degree n-1 with n terms. The term of degree k in P(n-1,x) has a coefficient given by T(n-1,k). Example: The third derivative of exp(1/x) is (exp(1/x)/x^6)*(1+6x+6x^2) and the 3rd row of this triangle is 1, 6, 6, which corresponds to this coefficients of the polynomial 1+6x+6x^2. - Derek Orr, Nov 06 2014
For another context for this array see the Callan (2008) article. -  Ron L.J. van den Burg, Dec 12 2021

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

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 203.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
David Callan, Sets, Lists and Noncrossing Partitions, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.3. Also on arXiv, arXiv:0711.4841 [math.CO], 2007-2008.
Tom Copeland, Lagrange a la Lah, 2011.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015
Olexandr Ganyushkin and Volodymyr Mazorchuk, Combinatorics of nilpotents in symmetric inverse semigroups, Ann. Comb. 8 (2004), no. 2, 161--175. [From _Abdullahi Umar_, Sep 14 2008]
F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
Matthieu Josuat-Vergès, Stammering tableaux - Tableaux bégayants, arXiv:1601.02212 [math.CO], 2016. See Lemma 7.1 p. 16.
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.
Jair Taylor, Number of acyclic digraphs on [n] with k edges and each indegree, outdegree <=1 (question on StackExchange)
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A000262 (row sums), A008297, A066667, A144084, row mirror of A105278.

P := n -> simplify(hypergeom([-n,-n+1],[],1/t));
seq(print(seq(coeff(expand(t^k*P(k)),t,k-j+1),j=1..k)),k=1..n); # Peter Luschny, Oct 29 2014

Table[(Binomial[n - 1, k - 1] Binomial[n, k - 1]/k) k!, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jul 04 2016 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1((n+1)!*binomial(n,k)/(n-k+1)!, ", ");); print(););} \\ Michel Marcus, Jan 12 2016

T(n, k) = A001263(n, k)*k!; A001263 = triangle of Narayana.
T(n, k) = C(n, n-k+1)*(n-1)!/(n-k)! = Sum_{i=n-k+1..n} |S1(n, i)*S2(i, n-k+1)| , with S1, S2 the Stirling numbers.
From Derek Orr, Mar 12 2015: (Start)
Each row represents a polynomial:
P(1,x) = 1;
P(2,x) = 1 + 2x;
P(3,x) = 1 + 6x + 6x^2;
P(4,x) = 1 + 12x + 36x^2 + 24x^3;
...
They are related through P(n+1,x) = x^2*P'(n,x) - (1+2*n*x)*P(n,x) with P(1,x) = 1.
(End)
From Peter Bala, Jul 04 2016: (Start)
Working with an offset of 0:
G.f.: exp(x*t)*I_1(2*sqrt(x)) = 1 + (1 + 2*t)*x/(1!*2!) + (1 + 6*t + 6*t^2)*x^2/(2!*3!) + (1 + 12*t + 36*t^2 + 24*t^3)*x^3/(3!*4!) + ..., where I_1(x) = Sum_{n >= 0} (x/2)^(2*n)/(n!*(n+1)!) is a modified Bessel function of the first kind.
The row polynomials R(n,t) satisfy R(n,t + u) = Sum_{k = 0..n} T(n,k)*t^k*R(n-k,u).
R(n,t) = 1 + Sum_{k = 0..n-1} (-1)^(n-k+1)*(n+1)!/(k+1)!* binomial(n,k)*t^(n-k)*R(k,t). Cf. A144084. (End)
From Peter Bala, Oct 05 2019: (Start)
The following formulas use a column index k starting at 0:
E.g.f.: exp(x/(1 - t*x)) - 1 = x + (1 + 2*t)*x^2/2! + (1 + 6*t + 6*t^2)*x^3/3! + ....
Recurrence for row polynomials: R(n+1,t) = (1 + 2*n*t)R(n,t) - n*(n-1)*t^2*R(n-1,t), with R(1,t) = 1 and R(2,t) = 1 + 2*t.
R(n+1,t) equals the numerator polynomial of the finite continued fraction 1 + n*t/(1 + n*t/(1 + (n-1)*t/(1 + (n-1)*t/(1 + ... + 2*t/(1 + 2*t/(1 + t/(1 + t/(1)))))))). The denominator polynomial is the n-th row polynomial of A144084. (End)
T(n,k) = A105278(n,n-k). - Ron L.J. van den Burg, Dec 12 2021

StackExchange link added by Felix A. Pahl, Dec 25 2012

A066668 Signed row sums of A066667.

Original entry on oeis.org

1, 1, 1, -1, -19, -151, -1091, -7841, -56519, -396271, -2442439, -7701409, 145269541, 4833158329, 104056218421, 2002667085119, 37109187217649, 679877731030049, 12440309297451121, 227773259993414719, 4155839606711748061, 74724654677947488521, 1293162252850914402221

Offset: 0

Christian G. Bower, Dec 17 2001

sign

Numerators in exp(x/(x+1)) power series (signs are different). - Benoit Cloitre, Mar 13 2002

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)=j-i if j>i. - Vladeta Jovovic, Jan 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A111884.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(x-1))/(1-x)^2)); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2018

a := n -> n!*hypergeom([1-n], [2], 1):
seq(simplify(a(n)), n=1..19); # Peter Luschny, Mar 30 2015

CoefficientList[Series[E^(x/(x-1))/(1-x)^2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 13 2014 *)
Table[Sum[-BellY[n+1, k, -Range[n+1]!], {k, n+1}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

x='x+O('x^30); Vec(serlaplace(exp(x/(x-1))/(1-x)^2)) \\ G. C. Greubel, May 15 2018

A066668 = lambda n: (-1)^n*hypergeometric([-n-1,-n-1,-n],[-n-1],-1)
[Integer(A066668(n).n(100)) for n in range(23)] # Peter Luschny, Sep 22 2014

a(n) = n!LaguerreL(n, 1, 1). - Paul Barry, Sep 08 2004
E.g.f.: exp(x/(x-1))/(1-x)^2.
Conjecture: a(n) +(-2*n+1)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
E.g.f. with a different offset: 1 - product {n >= 1} (1 - x^n)^(phi(n)/n) = x + x^2/2 + x^3/6 - x^4/24 - 19*x^5/120 - ..., where phi(n) = A000010(n) is the Euler totient function. Cf. A000262. - Peter Bala, Jan 01 2014
a(n) = (-1)^n*hypergeom([-n-1,-n-1,-n],[-n-1],-1). - Peter Luschny, Sep 22 2014
a(n) = n!*hypergeom([1-n], [2], 1). - Peter Luschny, Mar 30 2015

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

A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

Original entry on oeis.org

1, 1, -1, 2, -4, 1, 6, -18, 9, -1, 24, -96, 72, -16, 1, 120, -600, 600, -200, 25, -1, 720, -4320, 5400, -2400, 450, -36, 1, 5040, -35280, 52920, -29400, 7350, -882, 49, -1, 40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1, 362880, -3265920

Offset: 0

N. J. A. Sloane

In absolute values, this sequence also gives the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals (j-1)^2 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006
A partial permutation on a set X is a bijection between two subsets of X. |T(n,n-k)| equals the numbers of partial permutations of an n-set having domain cardinality equal to k. Let E denote the operator D*x*D, where D is the derivative operator d/dx. Then E^n = Sum_{k = 0..n} |T(n,k)|*x^k*D^(n+k). - Peter Bala, Oct 28 2008
The unsigned triangle is the generalized Riordan array (exp(x), x) with respect to the sequence n!^2 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!*(n+1)! is A105278). - Peter Bala, Aug 15 2013
The unsigned triangle appears on page 83 of Ser (1933). - N. J. A. Sloane, Jan 16 2020

			The triangle a(n,m) starts:
n\m   0       1      2       3      4      5    6  7  8
0:    1
1:    1      -1
2:    2      -4      1
3:    6     -18      9      -1
4:   24     -96     72     -16      1
5:  120    -600    600    -200     25     -1
6:  720   -4320   5400   -2400    450    -36    1
7: 5040  -35280  52920  -29400   7350   -882   49  -1
8:40320 -322560 564480 -376320 117600 -18816 1568 -64 1
...
From _Wolfdieter Lang_, Jan 31 2013 (Start)
Recurrence (usual one): a(4,1) = 7*(-18) - 6 - 3^2*(-4) = -96.
Recurrence (simplified version): a(4,1) = 5*(-18) - 6 = -96.
Recurrence (Sage program): |a(4,1)| = 6 + 3*18 + 4*9 = 96. (End)
Embedded recurrence (Maple program): a(4,1) = -4!*(1 + 3) = -96.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 83.

T. D. Noe, Rows n = 0..50 of triangle, flattened
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].
W. A. Al-Salam, Operational representations for the Laguerre and other polynomials, Duke Math. Jour., vol 31 (1964), pp. 127-142.
Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 5.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 7.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 21.
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.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
A. Belov-Kanel and M. Kontsevich, Automorphisms of the Weyl algebra, arXiv preprint arXiv:0512169 [math.QA], 2005.
A. Belov-Kanel and M. Kontsevich, The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture, arXiv preprint arXiv:0512171 [math.RA], 2005.
I. Gessel, Applications of the classical umbral calculus
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See. p. 19.
Massimo Nocentini, An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation, PhD Thesis, University of Florence, 2019. See p. 31.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Index entries for sequences related to Laguerre polynomials

Row sums give A009940, alternating row sums are A002720.
Column sequences (unsigned): A000142, A001563, A001809-A001812 for m=0..5.
Central terms: A295383.
For generators and generalizations see A132440.
Cf. A021010, A025166, A025167, A062137, A062138, A062139, A062140, A066667, A321620.

/* As triangle: */ [[((-1)^k)*Factorial(n)*Binomial(n, k)/Factorial(k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 18 2020

A021009 := proc(n,k) local S; S := proc(n,k) option remember; `if`(k = 0, 1, `if`( k > n, 0, S(n-1,k-1)/k + S(n-1,k))) end: (-1)^k*n!*S(n,k) end: seq(seq(A021009(n,k), k=0..n), n=0..8); # Peter Luschny, Jun 21 2017
# Alternative for the unsigned case (function RiordanSquare defined in A321620):
RiordanSquare(add(x^m, m=0..10), 10, true); # Peter Luschny, Dec 06 2018

Flatten[ Table[ CoefficientList[ n!*LaguerreL[n, x], x], {n, 0, 9}]] (* Jean-François Alcover, Dec 13 2011 *)

p(n) = denominator(bestapprPade(Ser(vector(2*n, k, (k-1)!))));
concat(1, concat(vector(9, n, Vec(-p(n)))))  \\ Gheorghe Coserea, Dec 01 2016

{T(n, k) = if( n<0, 0, n! * polcoeff( sum(i=0, n, binomial(n, n-i) * (-x)^i / i!), k))}; /* Michael Somos, Dec 01 2016 */

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

def A021009_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*k+1)*M[n-1,k]+(k+1)^2*M[n-1,k+1]
    return M
A021009_triangle(9) # Peter Luschny, Sep 19 2012

a(n, m) = ((-1)^m)*n!*binomial(n, m)/m! = ((-1)^m)*((n!/m!)^2)/(n-m)! if n >= m, otherwise 0.
E.g.f. for m-th column: (-x/(1-x))^m /((1-x)*m!), m >= 0.
Representation (of unsigned a(n, m)) as special values of Gauss hypergeometric function 2F1, in Maple notation: n!*(-1)^m*hypergeom([ -m, n+1 ], [ 1 ], 1)/m!. - Karol A. Penson, Oct 02 2003
Sum_{m>=0} (-1)^m*a(n, m) = A002720(n). - Philippe Deléham, Mar 10 2004
E.g.f.: (1/(1-x))*exp(x*y/(x-1)). - Vladeta Jovovic, Apr 07 2005
Sum_{n>=0, m>=0} a(n, m)*(x^n/n!^2)*y^m = exp(x)*BesselJ(0, 2*sqrt(x*y)). - Vladeta Jovovic, Apr 07 2005
Matrix square yields the identity matrix: L^2 = I. - Paul D. Hanna, Nov 22 2008
From Tom Copeland, Oct 20 2012: (Start)
Symbolically, with D=d/dx and LN(n,x)=n!L_n(x), define :Dx:^j = D^j x^j, :xD:^j = x^j D^j, and LN(.,x)^j = LN(j,x) = row polynomials of A021009.
Then some useful relations are
1) (:Dx:)^n = LN(n,-:xD:)    [Rodriguez formula]
2) (xDx)^n = x^n D^n x^n = x^n LN(n,-:xD:)  [See Al-Salam ref./A132440]
3) (DxD)^n = D^n x^n D^n = LN(n,-:xD:) D^n  [See ref. in A132440]
4) umbral composition LN(n,LN(.,x))= x^n   [See Rota ref.]
5) umbral comp. LN(n,-:Dx:) = LN(n,-LN(.,-:xD:)) = 2^n LN(n,-:xD:/2)= n! * (n-th row e.g.f.(x) of A038207 with x replaced by :xD:).
An example for 2) is the operator (xDx)^2 = (xDx)(xDx) = xD(x^2 + x^3D)= 2x^2 + 4x^3 D + x^4 D^2 = x^2 (2 + 4x D + x^2 D^2) = x^2 (2 + 4 :xD: + :xD:^2) = x^2 LN(2,-:xD:) = x^2 2! L_2(-:xD:).
An example of the umbral composition in 5) is given in A038207.
The op. xDx is related to the Euler/binomial transformation for power series/o.g.f.s. through exp(t*xDx) f(x) = f[x/(1-t*x)]/(1-t*x) and to the special Moebius/linear fractional/projective transformation z exp(-t*zDz)(1/z)f(z) = f(z/(1+t*z)).
For a general discussion of umbral calculus see the Gessel link. (End)
From Wolfdieter Lang, Jan 31 2013: (Start)
Standard recurrence derived from the three term recurrence of the orthogonal polynomials system {n!*L(n,x)}: L(n,x) = (2*n  - 1 - x)*L(n-1,x) - (n-1)^2*L(n-2,x), n>=1, L(-1,x) = 0, L(0,x) = 1.
  a(n,m) = (2*n-1)*a(n-1,m) - a(n-1,m-1) - (n-1)^2*a(n-2,m),
  n >=1, with a(n,-1) = 0, a(0,0) = 1, a(n,m) = 0 if n < m. (compare this with Peter Luschny's program for the unsigned case |a(n,m)| = (-1)^m*a(n,m)).
Simplified recurrence (using column recurrence from explicit form for a(n,m) given above):
a(n,m) = (n+m)*a(n-1,m) - a(n-1,m-1), n >= 1, a(0,0) = 1, a(n,-1) = 0, a(n,m) = 0 if n < m. (End)
|T(n,k)| = [x^k] (-1)^n*U(-n,1,-x), where U(a,b,x) is Kummer's hypergeometric U function. - Peter Luschny, Apr 11 2015
T(n,k) = (-1)^k*n!*S(n,k) where S(n,k) is recursively defined by: "if k = 0 then 1 else if k > n then 0 else S(n-1,k-1)/k + S(n-1,k)". - Peter Luschny, Jun 21 2017
The unsigned case is the exponential Riordan square (see A321620) of the factorial numbers. - Peter Luschny, Dec 06 2018
Omitting the diagonal and signs, this array is generated by the commutator [D^n,x^n] = D^n x^n - x^n D^n = Sum_{i=0..n-1} ((n!/i!)^2/(n-i)!) x^i D^i on p. 9 of both papers by Belov-Kanel and Kontsevich. - Tom Copeland, Jan 23 2020

Name changed and table given by Wolfdieter Lang, Nov 28 2011

A001705 Generalized Stirling numbers: a(n) = n! * Sum_{k=0..n-1} (k+1)/(n-k).

Original entry on oeis.org

0, 1, 5, 26, 154, 1044, 8028, 69264, 663696, 6999840, 80627040, 1007441280, 13575738240, 196287356160, 3031488633600, 49811492505600, 867718162483200, 15974614352793600, 309920046408806400, 6320046028584960000, 135153868608460800000, 3024476051557847040000

Offset: 0

N. J. A. Sloane

a(n) is also the sum of the positions of the right-to-left minima in all permutations of [n]. Example: a(3)=26 because the positions of the right-to-left minima in the permutations 123,132,213,231,312 and 321 are 123, 13, 23, 3, 23 and 3, respectively and 1 + 2 + 3 + 1 + 3 + 2 + 3 + 3 + 2 + 3 + 3 = 26. - Emeric Deutsch, Sep 22 2008
The asymptotic expansion of the higher order exponential integral E(x,m=2,n=2) ~ exp(-x)/x^2*(1 - 5/x + 26/x^2 - 154/x^3 + 1044/x^4 - 8028/x^5 + 69264/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009
a(n) is the total number of cycles (excluding fixed points) in all permutations of [n+1]. - Olivier Gérard, Oct 23 2012; Dec 31 2012
A length n sequence is formed by randomly selecting (one-by-one) n real numbers in (0,1).  a(n)/(n+1)! is the expected value of the sum of the new maximums in such a sequence.  For example for n=3: If we select (in this order): 0.591996, 0.646474, 0.163659  we would add 0.591996 + 0.646474 which would be a bit above the average of a(3)/4! = 26/24. - Geoffrey Critzer, Oct 17 2013

			(1-x)^-2 * (-log(1-x)) = x + 5/2*x^2 + 13/3*x^3 + 77/12*x^4 + ...
Examples: a(6) = 6!*(1/6 + 2/5 + 3/4 + 4/3 + 5/2 + 6/1) = 8028; a(20) = 20!*(1/20 + 2/19 + 3/18 + 4/17 + 5/16 + ... + 16/5 + 17/4 + 18/3 + 19/2 + 20/1) = 135153868608460800000. - _Alexander Adamchuk_, Oct 09 2004
From _Olivier Gérard_, Dec 31 2012: (Start)
The cycle decomposition of all permutations of 4 elements gives the following list: {{{1},{2},{3},{4}}, {{1},{2},{3,4}}, {{1},{2,3},{4}}, {{1},{2,4,3}}, {{1},{2,3,4}}, {{1},{2,4},{3}}, {{1,2},{3},{4}}, {{1,2},{3,4}}, {{1,3,2},{4}},{{1,4,3,2}}, {{1,3,4,2}}, {{1,4,2},{3}}, {{1,2,3},{4}}, {{1,2,4,3}},{{1,3},{2},{4}}, {{1,4,3},{2}}, {{1,3},{2,4}}, {{1,4,2,3}}, {{1,2,3,4}}, {{1,2,4},{3}}, {{1,3,4},{2}}, {{1,4},{2},{3}}, {{1,3,2,4}}, {{1,4},{2,3}}}.
Deleting the fixed points gives the following 26 items: {{3,4}, {2,3}, {2,4,3}, {2,3,4}, {2,4}, {1,2}, {1,2}, {3,4}, {1,3,2}, {1,4,3,2}, {1,3,4,2}, {1,4,2}, {1,2,3}, {1,2,4,3}, {1,3}, {1,4,3}, {1,3}, {2,4}, {1,4,2,3}, {1,2,3,4}, {1,2,4}, {1,3,4}, {1,4}, {1,3,2,4}, {1,4}, {2,3}}. (End)

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).

G. C. Greubel, Table of n, a(n) for n = 0..445 (terms 0 through 100 were provided by T. D. Noe)
J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, preprint, 2016.
J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, 340(10) (2017), 2550-2558.
Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021.
Gang Chen, Henrik Johansson, Fei Teng, and Tianheng Wang, Next-to-MHV Yang-Mills kinematic algebra, arXiv:2104.12726 [hep-th], 2021. See p. 46.
Loïc Foissy, Cointeraction on noncrossing partitions and related polynomial invariants, arXiv:2501.18212 [math.CO], 2025. See p. 24.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 406.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49. [Annotated scanned copy]
Motohico Mulase, In Search of a Hidden Curve, arXiv:2501.00716 [math.QA], 2025. See p. 27.
J. Riordan, Letter of 04/11/74.
J. Riordan, Letter, Jul 06 1978.

Cf. A000254 (total number of cycles in permutations, including fixed points).
Cf. A002104 (number of different cycles in permutations, without fixed points).
Cf. A006231 (number of different cycles in permutations, including fixed points).
Cf. A001008, A002805, A006675, A066667, A132159, A143947.
Related to n!*the k-th successive summation of the harmonic numbers:
  (k=0) A000254, (k=1) A001705, (k=2) A001711, (k=3) A001716,
  (k=4) A001721, (k=5) A051524, (k=6) A051545, (k=7) A051560,
  (k=8) A051562, (k=9) A051564.

a := n-> add((n+1)!/k, k=2..n+1): seq(a(n), n=0..21); # Zerinvary Lajos, Jan 22 2008; edited Johannes W. Meijer, Nov 28 2012
a := n -> ((n+1)!*(h(n+1)-1)): h := n-> harmonic(n): seq(a(n), n=0..21); # Gary Detlefs, Dec 18 2009; corrected by Johannes W. Meijer, Nov 28 2012

Table[n!*Sum[Sum[1/k,{k,1,m}], {m,1,n}], {n,0,20}] (* Alexander Adamchuk, Apr 14 2006 *)
a[n_] := (n + 1)! (EulerGamma - 1 + PolyGamma[n + 2]);
Table[a[n], {n, 0, 21}] (* Peter Luschny, Feb 19 2022 *)

a(n):=n!*sum(((-1)^(k+1)*binomial(n+1,k+1))/k,k,1,n); /* Vladimir Kruchinin, Oct 10 2016 */

for(n=0,25, print1(n!*sum(k=0,n-1,(k+1)/(n-k)), ", ")) \\ G. C. Greubel, Jan 20 2017

from math import factorial
def A001705(n):
    f = factorial(n)
    return sum(f*(k+1)//(n-k) for k in range(n)) # Chai Wah Wu, Jun 23 2022

Partial sum of first n harmonic numbers multiplied by n!.
a(n) = n!*Sum_{m=1..n} Sum_{k=1..m} 1/k = n!*Sum_{m=1..n} H(m), where H(m) = Sum_{k=1..m} 1/k = A001008(m)/A002805(m) is m-th Harmonic number.
E.g.f.: - log (1 - x) / (1 - x)^2.
a(n) = (n+1)! * H(n) - n*n!, H(n) = Sum_{k=1..n} (1/k).
a(n) = A112486(n, 1).
a(n) = a(n-1)*(n+1) + n! = A000254(n+1) - A000142(n+1) = A067176(n+1, 1). - Henry Bottomley, Jan 09 2002
a(n) = Sum_{k=0..n-1} ((-1)^(n-1+k) * (k+1) * 2^k * Stirling1(n, k+1)). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at 0. - Ralf Stephan, Apr 16 2004
a(n) = Sum_{k=1..n} (k*StirlingCycle(n+1,k+1)). - David Callan, Sep 25 2006
a(n) = Sum_{k=n..n*(n+1)/2} k*A143947(n,k). - Emeric Deutsch, Sep 22 2008
For n >= 1, a(n) = Sum_{j=0..n-1} ((-1)^(n-j-1) * 2^j * (j+1) * Stirling1(n,j+1)). - Milan Janjic, Dec 14 2008
a(n) = (2*n+1)*a(n-1) - n^2*a(n-2). - Gary Detlefs, Nov 27 2009
a(n) = (n+1)!*(H(n+1) - 1) where H(n) is the n-th harmonic number. - Gary Detlefs, Dec 18 2009
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n+1,k+1)/k. - Vladimir Kruchinin, Oct 10 2016
a(n) = (n+1)!*Sum_{k = 1..n} (-1)^(k+1)*binomial(n+1,k+1)*k/(k+1). - Peter Bala, Feb 15 2022
a(n) = Gamma(n + 2) * (Digamma(n + 2) + EulerGamma - 1). - Peter Luschny, Feb 19 2022
From Mélika Tebni, Jun 22 2022: (Start)
a(n) = -Sum_{k=0..n} k!*A066667(n, k+1).
a(n) = Sum_{k=0..n} k!*A132159(n, k+1). (End)
a(n) = n*(n + 1)!*hypergeom([1, 1, 1 - n], [2, 3], 1)/2. - Peter Luschny, Jun 22 2022

More terms from Sascha Kurz, Mar 22 2002

A187535 Central Lah numbers: a(n) = A105278(2n,n) = A008297(2n,n).

Original entry on oeis.org

1, 2, 36, 1200, 58800, 3810240, 307359360, 29682132480, 3339239904000, 428906814336000, 61934143990118400, 9931984545324441600, 1751339941492209868800, 336796142594655744000000, 70149825129001153536000000, 15732267448930658699673600000

Offset: 0

Emanuele Munarini, Mar 11 2011

a(n) is the number of Lah partitions of a set of size 2n with n blocks.

Cf. A008297, A111596, A066667, A187536, A187538, A187539, A187540, A187542 - A187548, A090181, A125558.

A187535:= n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(A187535(n),n=0..12);

a[n_]:=If[n==0,1,Binomial[2n-1,n-1](2n)!/n!]
Table[a[n],{n,0,12}]
(* Alternative: *)
a[n_] := Binomial[2*n, n] FactorialPower[2*n - 1, n];
Table[a[n], {n, 0, 15}] (* Peter Luschny, Jun 15 2022 *)

a(n) := if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(a(n),n,0,12);

[catalan_number(n)*binomial(2*n-1,n)*factorial(n+1) for n in range(15)] # Peter Luschny, Oct 07 2014

a(n) = binomial(2n-1,n-1)*(2n)!/n! (for n>0).
D-finite with recurrence (n+1)*a(n+1) = 4*(2n+1)^2*a(n) - delta(n,0).
a(n) ~ 2^(4*n)*n^n*exp(-n)/sqrt(2*n*Pi).
a(n)*a(n+2) - a(n+1)^2 is >= 0 and is a multiple of 2^(n+3) for all nonnegative  n.
a(n) == 0 (mod 10) for n>3.
E.g.f.: 1/2 + K(16x)/Pi, where K(z) is the complete elliptic integral of the first kind, which can also be written as a Legendre function of the second kind.
a(n) = Catalan(n)*C(2*n-1,n)*(n+1)!. - Peter Luschny, Oct 07 2014
a(n) = A125558(n)*(n+1)! = A090181(2*n,n)*(n+1)!. - Peter Luschny, Oct 07 2014
a(n) = (2/n)*(Gamma(2*n)^2/Gamma(n)^3) for n>0. - Peter Luschny, Oct 17 2014

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

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Jun 19 2001

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

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

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

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

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

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

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

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

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

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

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

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

A062140 Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x).

Original entry on oeis.org

1, 5, -1, 30, -12, 1, 210, -126, 21, -1, 1680, -1344, 336, -32, 1, 15120, -15120, 5040, -720, 45, -1, 151200, -181440, 75600, -14400, 1350, -60, 1, 1663200, -2328480, 1164240, -277200, 34650, -2310, 77, -1, 19958400, -31933440

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,4,x)= sum(a(n,m)*x^m,m=0..n) have g.f. exp(-z*x/(1-z))/(1-z)^5. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(|A008297(n,m)|*(-x)^m, m=1..n) and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

			Triangle begins:
  {1};
  {5,-1};
  {30,-12,1};
  {210,-126,21,-1};
  ...
2!*L(2,4,x)=30-12*x+x^2.

Indranil Ghosh, Rows 0..125
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001720(n+4), A062199, A062260-A062263. The row sums (signed) give A062265, the row sums (unsigned) give A062266.

Cf. A021009, A062137-A062139, A066667.

Flatten[Table[((-1)^m)*n!*Binomial[n+4,n-m]/m!,{n,0,11},{m,0,n}]] (* Indranil Ghosh, Feb 23 2017 *)

row(n) = Vecrev(n!*pollaguerre(n, 4)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n,r):
    return f(n)//f(r)//f(n-r)
i=0
for n in range(26):
    for m in range(n+1):
        print(i, (-1)**m*f(n)*C(n+4,n-m)//f(m))
        i+=1 # Indranil Ghosh, Feb 23 2017

T(n, m) = ((-1)^m)*n!*binomial(n+4, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^5), m >= 0.

A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).

Original entry on oeis.org

1, 6, -1, 42, -14, 1, 336, -168, 24, -1, 3024, -2016, 432, -36, 1, 30240, -25200, 7200, -900, 50, -1, 332640, -332640, 118800, -19800, 1650, -66, 1, 3991680, -4656960, 1995840, -415800, 46200, -2772, 84, -1, 51891840, -69189120

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,5,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^6. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials sum(|A008297(n,m)|*(-x)^m, m=1..n), n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
These polynomials appear in the radial part of the l=2 (d-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148-A062152. Row sums (signed) give A062191; row sums (unsigned) give A062192.
The unsigned version of this triangle is the triangle of unsigned 3-Lah numbers A143498. - Peter Bala, Aug 25 2008

			Triangle begins:
  {1};
  {6, -1};
  {42, -14, 1};
  {336, -168, 24, -1};
  ...
2!*L(2, 5, x) = 42-14*x+x^2.

A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.

Indranil Ghosh, Rows 0..125, flattened
Index entries for sequences related to Laguerre polynomials

Cf. A021009, A062137, A062139, A062140, A066667.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148, A062149, A062150, A062151, A062152.
Row sums (signed) give A062191, row sums (unsigned) give A062192.
Cf. A143498.

Flatten[Table[((-1)^m)*n!*Binomial[n+5,n-m]/m!,{n,0,8},{m,0,n}]] (* Indranil Ghosh, Feb 24 2017 *)

tabl(nn) = {for (n=0, nn, for (m=0, n, print1(((-1)^m)*n!*binomial(n+5, n-m)/m!, ", "); ); print(); ); } \\ Indranil Ghosh, Feb 24 2017

row(n) = Vecrev(n!*pollaguerre(n, 5)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n, r):return f(n)//f(r)//f(n-r)
i=-1
for n in range(26):
    for m in range(n+1):
        i += 1
        print(str(i)+" "+str(((-1)**m)*f(n)*C(n+5, n-m)//f(m))) # Indranil Ghosh, Feb 24 2017

T(n, m) = ((-1)^m)*n!*binomial(n+5, n-m)/m!.

E.g.f. for m-th column: ((-x/(1-x))^m)/(m!*(1-x)^6), m >= 0.

A221913 Array of coefficients of numerator polynomials (divided by x) of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+...

Original entry on oeis.org

1, 2, 6, 1, 24, 6, 120, 36, 1, 720, 240, 12, 5040, 1800, 120, 1, 40320, 15120, 1200, 20, 362880, 141120, 12600, 300, 1, 3628800, 1451520, 141120, 4200, 30, 39916800, 16329600, 1693440, 58800, 630, 1, 479001600, 199584000, 21772800, 846720, 11760, 42

Offset: 1

Wolfdieter Lang, Feb 23 2013

The row length sequence of this array is 1 + floor((n-1)/2) = A008619(n-1), n >= 1.
The array of denominators is found under A084950.
The continued fraction 0 + K_{k=1..infinity}(x/k) = x/(1+x/(2+x/(3+... has n-th approximation P(n,x)/Q(n,x). These polynomials satisfy the recurrence q(n,x) = n*q(n-1,x) + x*q(n-2,x), for q replaced by P or Q with inputs P(-1,x) = 1, P(0,x) = 0 and Q(-1,x) = 0 and Q(0,1) = 1. The present array provides the coefficients for Phat(n,x) := P(n,x)/x = sum(a(n,m)*x^m,m=0..floor((n-1)/2)), n >= 1. The recurrence is that of q(n,x) and the inputs are Phat(-1,x) = 1/x and Phat(0,x) =0. For the Q(n,x) coefficients see the companion array A084950. The solution with input q(-1,x) = a and q(0,x) = b is then, due to linearity, q(a,b;n,x) = a*x*Phat(n,x) + b*Q(n,x).  The motivation to consider the q(n,x) recurrence stems from e-mails from Gary Detlefs, who considered integer x and various inputs and gave explicit formulas.
This array coincides with the SW-NE diagonals of the coefficient array |A066667| or A105278 (taken with offset [0,0]) of the generalized Laguerre polynomials n!*L(1,n,x) (parameter alpha = 1).
The entries a(n,m) have a combinatorial interpretation in terms of certain so-called labeled Morse code polynomials using dots (length 1) and dashes (of length 2). a(n,m) is the number of possibilities to decorate the n-1 positions 2,...,n with m dashes, m from {0,1,...,floor((n-1)/2)}, and n-1-2*m dots. A dot at position k has a label k and each dash between two neighboring positions has a label x.  a(n,m) is the sum of these labeled Morse codes with m dashes after the label x^m has been divided out.  E.g., a(6,2) = 6 + 4 + 2 = 12  from the 3 codes:  dash dash dot, dash dot dash,and  dot dash dash, or (23)(45)6, (23)4(56) and 2(34)(56), and labels (which are in general multiplicative)  6*x^2, 4*x^2 and 2*x^2, respectively.
For general Morse code polynomials (Euler's continuants) see the Graham et al. reference given in A221915, p. 302. - Wolfdieter Lang, Feb 28 2013
Row sums Phat(n,1) = A001053(n+1), n >= 1. Alternating row sums  Phat(n,-1) = A058798(n), n >= 1.
From Wolfdieter Lang, Mar 06 2013 (Start)
The recurrence for q(n,x) given above, can be transformed to the one of Bessel functions given in Abramowitz-Stegun (see A103921 for the reference) in the first line of eq. 9.1.27 on p. 361 via i^n*q(n,x)/sqrt(x)^n = C(n+1,-i*2*sqrt(x)) with the imaginary unit i, where C can stand for BesselJ or BesselY. In order to fix the two inputs for the Q or Phat polynomials (given above) one uses a linear combination of these two independent solutions. The Wronskian eq. 9.1.16, p. 360, is used to simplify the coefficients. One can also use an alternative version based on eqs. 9.6.3 and 9.6.5, p. 375, to trade the J and Y polynomials for I and K.
  This produces the two explicit formulas given below, and also the two versions given for Q in A084950.
(End)
For large order n the behavior of the row polynomials Phat(n,x) (see above) is known from the one of Bessel functions. See a comment on asymptotics under A084950. This leads then to the limit for Phat(n,x)/n! given in the formula section. The limit for the continued fraction mentioned in the name and above is also found in this comment on A084950. - Wolfdieter Lang, Mar 08 2013
This is the unsigned Lah triangle read by ascending antidiagonals. Conversely, reading the given triangle beginning at the left in descending steps yields a row of the unsigned Lah triangle. This can be verified immediately by means of the explicit formulas. For example, [T(5,0), T(6,1), T(7,2), T(8,3), T(9,4)] is row 5 of A105278. - Peter Luschny, Dec 07 2019

			The irregular triangle a(n,m)  begins:
n\m          0          1         2        3      4    5  6
1:           1
2:           2
3:           6          1
4:          24          6
5:         120         36         1
6:         720        240        12
7:        5040       1800       120        1
8:       40320      15120      1200       20
9:      362880     141120     12600      300     1
10:    3628800    1451520    141120     4200    30
11:   39916800   16329600   1693440    58800    63     1
12:  479001600   19958400  21772800   846720  11760   42
13: 6227020800 2634508800 299376000 12700800 211680 1176  1
...
Recurrence (short version): a(6,1) = 6*36 + 24 = 240.
Recurrence (long version): a(6,1) = 2*4*36 + 24 - 4*3*6 = 240.
a(6,1) = binomial(4,1)*5!/2! = 4*3*4*5 = 240.

Cf. |A066667|, A105278, A084950, A221915.

row[n_] := x/ContinuedFractionK[x, i, {i, 0, n}] // Simplify // Together // Numerator // CoefficientList[#, x]& // Rest;
row /@ Range[12] // Flatten (* Jean-François Alcover, Oct 28 2019 *)

Recurrence (short version): a(n,m) = n*a(n-1,m)  + a(n-2,m-1), n>=2, a(1,1) =1, a(n,-1) = 0, a(n,m) = 0 if n < 2*m+1. From the recurrence for the Phat(n,x) polynomials given in a comment above.
Recurrence (long version): a(n,m) = 2*(n-1-m)*a(n-1,m) + a(n-2,m-1) - (n-1-m)*(n-2-m)*a(n-2,m), n >= 1, a(1,0) = 1, a(n,-1) = 0, a(n,m) = 0 if  n < 2*m + 1. From the recurrence for the unsigned  generalized Laguerre polynomial with parameter alpha = 1. This recurrence can be simplified to the preceding short version, because the following explicit form follows from the one for the generalized Laguerre coefficients (which, in turn, derives from the Rodrigues formula and the Leibniz rule). This proves the relation a(n,m) = |Lhat(1,n-1-m,m)|, with the coefficients |Lhat(1,n,m)| = |A066667(n,m)|  of the unsigned n!*L(1,n,x) Laguerre polynomials (parameter alpha = 1).
a(n,m) = binomial(n-1-m,m)*(n-m)!/(m+1)!, n >= 1, 0 <= m <= floor((n-1)/2).
For the e.g.f.s of the column sequences see A105278 (here with different offset, which could be obtained by integration).
E.g.f. for row polynomials  gPhat(z,x) := Sum_{z>=0} Phat(n,x)*z^n = Pi*(BesselJ(1, 2*i*sqrt(x)*sqrt(1-z))*BesselY(1, 2*i*sqrt(x)) - BesselY(1, (2*i)*sqrt(x)*sqrt(1-z))*BesselJ(1, 2*i*sqrt(x)))/sqrt(1-z) with Bessel functions and the imaginary unit i = sqrt(-1). Phat(0,x) = 0.
From Wolfdieter Lang, Mar 06 2013 (Start)
For the row polynomials one finds Phat(n,x) = Pi*(z/2)^n*(BesselY(1,z)* BesselJ(n+1,z) - BesselJ(1,z)*BesselY(n+1,z)) where z := -i*2*sqrt(x) and the i is the imaginary unit. An alternative form is Phat(n,x) = 2*(w/2)^n*(BesselI(1,w)*BesselK(n+1,w) + BesselK(1,w)*BesselI(n+1,w)*(-1)^(n+1)), n >= 1, where w := -2*sqrt(x). See a comment above for the derivation. (End)
Limit_{n -> oo} Phat(n,x)/n! = BesselI(1,2*sqrt(x))/sqrt(x). See a comment above. - Wolfdieter Lang, Mar 08 2013

cp's OEIS Frontend

A089231 Triangular array A066667 or A008297 unsigned and transposed.

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A066668 Signed row sums of A066667.

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A008297 Triangle of Lah numbers.

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A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

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A001705 Generalized Stirling numbers: a(n) = n! * Sum_{k=0..n-1} (k+1)/(n-k).

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A187535 Central Lah numbers: a(n) = A105278(2n,n) = A008297(2n,n).