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%I
%S -1,2,1,-6,-6,-1,24,36,12,1,-120,-240,-120,-20,-1,720,1800,1200,300,
%T 30,1,-5040,-15120,-12600,-4200,-630,-42,-1,40320,141120,141120,58800,
%U 11760,1176,56,1,-362880,-1451520,-1693440,-846720,-211680,-28224,-2016,-72,-1,3628800,16329600,21772800,12700800
%N Triangle of Lah numbers.
%C |a(n,k)| = number of partitions of {1,..,n} into k lists, where a list means an ordered subset.
%C 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
%C 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
%C 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
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D S. Covo, The moments of a compound Poisson process with exponential or centered normal jumps, J. Probab. Stat. Sci., 7 (2009), 91-100. [From Shai Covo (green355(AT)netvision.net.il), Feb 09 2010]
%D 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. http://math.pugetsound.edu/~mspivey/Exp.pdf.
%D D. Karp and E. Prilepkina, Generalized Stieltjes transforms: basic aspects, Arxiv preprint arXiv:1111.4271, 2011
%D D. Karp and E. Prilepkina, Generalized Stieltjes functions and their exact order, Journal of Classical Analysis Volume 1, Number 1 (2012), 53-74, http://files.ele-math.com/articles/jca-01-07.pdf. - From _N. J. A. Sloane_, Dec 25 2012
%D U. N. Katugampola, A new Fractional Derivative and its Mellin Transform, Arxiv preprint arXiv:1106.0965, 2011
%D U. N. Katugampola, Mellin Transforms of the Generalized Fractional Integrals and Derivatives, Arxiv preprint arXiv:1112.6031, 2011
%D D. E. Knuth, Convolution polynomials, The Mathematica J., 2.1 (1992) 67-78.
%D 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; http://www.mat.unisi.it/newsito/puma/public_html/23_2/mansour_schork.pdf
%D 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.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%D S. G. Williamson, Combinatorics for Computer Science, Computer Science Press, 1985; see p. 176.
%H T. D. Noe, <a href="/A008297/b008297.txt">Rows n=1..100 of triangle, flattened</a>
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem.</a>
%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F a(n, m) = (-1)^n*n!*A007318(n-1, m-1)/m!, n >= m >= 1.
%F 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.
%F 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.
%F 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_
%F If L_n(y)=Sum_{k=0..n} |a(n, k)|*y^k (a Lah polynomial) then e.g.f. for L_n(y) is exp(x*y/(1-x)) - _Vladeta Jovovic_, Jan 06 2001
%F E.g.f. for k-th column (unsigned): x^k/(1-x)^k/k!. - _Vladeta Jovovic_, Dec 03 2002
%F a(n, k) = (n-k+1)!*N(n, k) where N(n, k) is the Narayana triangle A001263. - _Philippe Deléham_, Jul 20 2003
%F Contribution from Shai Covo (green355(AT)netvision.net.il), Feb 02 2010: (Start)
%F 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):
%F 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;
%F 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;
%F 3) L_n(y)=(2n-2+y)L_{n-1}(y)-(n-1)(n-2)L_{n-2}(y), n>=2;
%F 4) L_n(y)= y*(n-1)!*Sum_{k=1..n} (L_{n-k}(y) k!)/((n-k)! (k-1)!), n>=1. (End)
%F 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
%F 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
%e |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).
%e Triangle:
%e -1
%e 2, 1
%e -6, -6, -1
%e 24, 36, 12, 1
%e -120, -240, -120, -20, -1 ...
%p A008297 := (n,m) -> (-1)^n*n!*binomial(n-1,m-1)/m!;
%t 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 *)
%o (Sage)
%o def A008297_triangle(dim): # computes unsigned T(n, k).
%o M = matrix(SR,dim,dim)
%o for n in (0..dim-1): M[n,n] = 1
%o for n in (1..dim-1):
%o for k in (0..n-1):
%o M[n,k] = M[n-1,k-1]+(2+2*k)*M[n-1,k]+((k+1)*(k+2))*M[n-1,k+1]
%o return M
%o A008297_triangle(9) # _Peter Luschny_, Sep 19 2012
%Y 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.
%Y Columns 1-6 (unsigned): A000142, A001286, A001754, A001755, A001777, A001778.
%Y Cf. A001263. A111596 (differently signed triangle with extra column m=0 and row n=0).
%Y Cf. A130561 for a natural refinement.
%K sign,tabl,nice,changed
%O 1,2
%A _N. J. A. Sloane_.
%E More terms from _James A. Sellers_, Jan 03 2001
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