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A008297 Triangle of Lah numbers. 105
-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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

|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

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.

S. Covo, The moments of a compound Poisson process with exponential or centered normal jumps, J. Probab. Stat. Sci., 7 (2009), 91-100.

T. 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.

Jose L. Ramirez, M Shattuck, A (p, q)-Analogue of the r-Whitney-Lah Numbers, Journal of Integer Sequences, 19, 2016, #16.5.6.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

S. G. Williamson, Combinatorics for Computer Science, Computer Science Press, 1985; see p. 176.

LINKS

T. D. Noe, Rows n=1..100 of triangle, flattened

J. Fernando Barbero G., Jesús Salas, 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.

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015

T. Copeland, Lagrange a la Lah , 2011.

S. Daboul, J. Mangaldan, M. Z. Spivey and P. Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., 86 (2013), 39-47.

A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [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, 22(4) (2015), #P4.10.

M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3

D. Karp and E. Prilepkina, Generalized Stieltjes transforms: basic aspects, arXiv preprint arXiv:1111.4271 [math.CA], 2011.

D. Karp and E. Prilepkina, Generalized Stieltjes functions and their exact order, Journal of Classical Analysis Volume 1, Number 1 (2012), 53-74. - N. J. A. Sloane, Dec 25 2012

U. N. Katugampola, A new Fractional Derivative and its Mellin Transform, arXiv preprint arXiv:1106.0965 [math.CA], 2011.

U. N. Katugampola, Mellin Transforms of the Generalized Fractional Integrals and Derivatives, arXiv preprint arXiv:1112.6031 [math.CA], 2011.

D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

T. Mansour and M. Schork, Generalized Bell numbers and algebraic differential equations, Pure Math. Appl.(PU. MA), Vol. 23 (2012), No. 2, pp. 131-142.

Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

FORMULA

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_(A008275(n, k)*A008277(k, m), k=m..n) where A008275 = positive 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)

EXAMPLE

|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 ...

MAPLE

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

MATHEMATICA

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 *)

PROG

(Sage)

def A008297_triangle(dim): # computes unsigned T(n, k).

    M = matrix(SR, 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

(Haskell)

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

(PARI) 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

(Perl) 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

CROSSREFS

Same as A066667 and A105278 except for signs. Cf. A007318, A048786. Row sums of unsigned triangle form A000262(n). A002868 gives maximal element (in magnitude) in each row.

Columns 1-6 (unsigned): A000142, A001286, A001754, A001755, A001777, A001778.

Cf. A001263. A111596 (differently signed triangle with extra column m=0 and row n=0).

Cf. A130561 for a natural refinement.

Cf. A248045 (central terms, negated).

Cf. A008275, A008277, A048993, A094638, A130534.

Sequence in context: A048999 A066667 A105278 * A090582 A079641 A222864

Adjacent sequences:  A008294 A008295 A008296 * A008298 A008299 A008300

KEYWORD

sign,tabl,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Jan 03 2001

STATUS

approved

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Last modified May 25 12:08 EDT 2017. Contains 287027 sequences.