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%I #197 Feb 02 2023 17:11:13
%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
%C 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
%C Named after the Slovenian mathematician and actuary Ivo Lah (1896-1979). - _Amiram Eldar_, Jun 13 2021
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D 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.
%D 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.
%D John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
%D S. Gill 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 J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013.
%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>, arXiv:quant-ph/0212072, 2002.
%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>, arXiv:quant-ph/0402027, 2004.
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015.
%H Tom Copeland, <a href="https://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>, 2011.
%H Siad Daboul, Jan Mangaldan, Michael Z. Spivey and Peter J. Taylor, <a href="http://math.pugetsound.edu/~mspivey/Exp.pdf">The Lah Numbers and the n-th Derivative of exp(1/x)</a>, Math. Mag., Vol. 86, No. 1 (2013), pp. 39-47.
%H Askar Dzhumadil'daev and Damir Yeliussizov, <a href="http://arxiv.org/abs/1408.6764v1">Path decompositions of digraphs and their applications to Weyl algebra</a>, 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]
%H Askar Dzhumadil’daev and Damir Yeliussizov, <a href="https://doi.org/10.37236/5181">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, Vol. 22, No. 4 (2015), #P4.10.
%H B. S. El-Desouky, Nenad P.Cakić and Toufik Mansour, <a href="https://doi.org/10.1016/j.aml.2009.08.018">Modified approach to generalized Stirling numbers via differential operators</a>, Appl. Math. Lett., Vol. 23, No. 1 (2010), pp. 115-120.
%H Sen-Peng Eu, Tung-Shan Fu, Yu-Chang Liang and Tsai-Lien Wong. <a href="https://arxiv.org/abs/1701.00600">On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks</a>. arXiv:1701.00600 [math.CO], 2017.
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS, Vol. 12 (2009), pp. 09.8.3
%H Dmitry Karp and Elena Prilepkina, <a href="http://arxiv.org/abs/1111.4271">Generalized Stieltjes transforms: basic aspects</a>, arXiv preprint arXiv:1111.4271 [math.CA], 2011.
%H Dmitry Karp and Elena Prilepkina, <a href="http://files.ele-math.com/articles/jca-01-07.pdf">Generalized Stieltjes functions and their exact order</a>, Journal of Classical Analysis, Vol. 1, No. 1 (2012), pp. 53-74. - _N. J. A. Sloane_, Dec 25 2012.
%H Udita N. Katugampola, <a href="http://arxiv.org/abs/1106.0965">A new Fractional Derivative and its Mellin Transform</a>, arXiv preprint arXiv:1106.0965 [math.CA], 2011.
%H Udita N. Katugampola, <a href="http://arxiv.org/abs/1112.6031">Mellin Transforms of the Generalized Fractional Integrals and Derivatives</a>, arXiv preprint arXiv:1112.6031 [math.CA], 2011.
%H Donald E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, Mathematica J. 2.1 (1992), no. 4, 67-78; arXiv:math/9207221 [math.CA], 1992.
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
%H Toufik Mansour and Matthias Schork, <a href="http://puma.dimai.unifi.it/23_2/mansour_schork.pdf">Generalized Bell numbers and algebraic differential equations</a>, Pure Math. Appl.(PU. MA), Vol. 23, No. 2 (2012), pp. 131-142.
%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.3.
%H Toufik Mansour and Mark Shattuck, <a href="https://dx.doi.org/10.1007/s10998-017-0209-9">A polynomial generalization of some associated sequences related to set partitions</a>, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
%H Emanuele Munarini, <a href="https://doi.org/10.2298/AADM180226017M">Combinatorial identities involving the central coefficients of a Sheffer matrix</a>, Applicable Analysis and Discrete Mathematics, Vol. 13, No. 2 (2019), pp. 495-517.
%H Mathias Pétréolle and Alan D. Sokal, <a href="https://arxiv.org/abs/1907.02645">Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions</a>, arXiv:1907.02645 [math.CO], 2019.
%H Jose L. Ramirez and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Ramirez2/ramirez12.pdf">A (p, q)-Analogue of the r-Whitney-Lah Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.6.
%H Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Shattuck/sha11.html">Identities for Generalized Whitney and Stirling Numbers</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.10.4.
%H Mark Shattuck, <a href="https://doi.org/10.33039/ami.2018.11.001">Some formulas for the restricted r-Lah numbers</a>, Annales Mathematicae et Informaticae, Vol. 49 (2018), Eszterházy Károly University Institute of Mathematics and Informatics, pp. 123-140.
%H Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Shattuck/sha27.html">Combinatorial Proofs of Some Stirling Number Convolution Formulas</a>, J. Int. Seq., Vol. 25 (2022), Article 22.2.2.
%H Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq., Vol. 14 (2011) Article 11.9.7.
%H Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%H Bao-Xuan Zhu, <a href="https://arxiv.org/abs/2006.14485">Total positivity from a generalized cycle index polynomial</a>, arXiv:2006.14485 [math.CO], 2020.
%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_{k=m..n} |A008275(n, k)|*A008277(k, m), where A008275 = 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 the e.g.f. for L_n(y) is exp(x*y/(1-x)). - _Vladeta Jovovic_, Jan 06 2001
%F E.g.f. for the 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 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
%F From _Tom Copeland_, Sep 25 2016: (Start)
%F 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.
%F 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.
%F 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).
%F 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)
%F T(n,k) = Sum_{j=k..n} (-1)^j*A008296(n,j)*A360177(j,k). - _Mélika Tebni_, Feb 02 2023
%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(ZZ,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
%o (Haskell)
%o a008297 n k = a008297_tabl !! (n-1) !! (k-1)
%o a008297_row n = a008297_tabl !! (n-1)
%o a008297_tabl = [-1] : f [-1] 2 where
%o f xs i = ys : f ys (i + 1) where
%o ys = map negate $
%o zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))
%o -- _Reinhard Zumkeller_, Sep 30 2014
%o (PARI) T(n, m) = (-1)^n*n!*binomial(n-1, m-1)/m!
%o for(n=1,9, for(m=1,n, print1(T(n,m)", "))) \\ _Charles R Greathouse IV_, Mar 09 2016
%o (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
%Y 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.
%Y A293125 (row sums) and A000262 (row sums of unsigned triangle).
%Y Columns 1-6 (unsigned): A000142, A001286, A001754, A001755, A001777, A001778.
%Y A002868 gives maximal element (in magnitude) in each row.
%Y A248045 (central terms, negated). A130561 is a natural refinement.
%Y Cf. A007318, A048786, A001263, A008275, A008277, A048993, A094638, A130534.
%Y Cf. A008296, A360177.
%K sign,tabl,nice,easy
%O 1,2
%A _N. J. A. Sloane_
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