Search: id:A008297
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%I A008297
%S A008297 1,2,1,6,6,1,24,36,12,1,120,240,120,20,1,720,1800,1200,300,30,1,5040,15120,
%T A008297 12600,4200,630,42,1,40320,141120,141120,58800,11760,1176,56,1,362880,
               1451520,
%U A008297 1693440,846720,211680,28224,2016,72,1,3628800,16329600,21772800,12700800
%V A008297 -1,2,1,-6,-6,-1,24,36,12,1,-120,-240,-120,-20,-1,720,1800,1200,300,30,
               1,-5040,-15120,
%W A008297 -12600,-4200,-630,-42,-1,40320,141120,141120,58800,11760,1176,56,1,-362880,
               -1451520,
%X A008297 -1693440,-846720,-211680,-28224,-2016,-72,-1,3628800,16329600,21772800,
               12700800
%N A008297 Triangle of Lah numbers.
%C A008297 |a(n,k)| = number of partitions of {1,..,n} into k lists, where a list 
               means an ordered subset.
%D A008297 T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.
%D A008297 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
%D A008297 D. E. Knuth, Convolution polynomials, The Mathematica J., 2.1 (1992) 
               67-78.
%D A008297 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+}.
%D A008297 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               44.
%D A008297 S. G. Williamson, Combinatorics for Computer Science, Computer Science 
               Press, 1985; see p. 176.
%H A008297 T. D. Noe, <a href="b008297.txt">Rows n=1..100 of triangle, flattened</
               a>
%H A008297 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 A008297 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 A008297 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 A008297 a(n, m) = (-1)^n*n!*A007318(n-1, m-1)/m!, n >= m >= 1.
%F A008297 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 A008297 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 A008297 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(AT)physik.uni-karlsruhe.de
%F A008297 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 (vladeta(AT)eunet.rs), 
               Jan 06 2001
%F A008297 E.g.f. for k-th column (unsigned): x^k/(1-x)^k/k!. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Dec 03 2002
%F A008297 a(n, k) = (n-k+1)!*N(n, k) where N(n, k) is the Narayana triangle AOO1263. 
               - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 20 2003
%e A008297 |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 A008297 -1; 2,1; -6,-6,-1; 24,36,12,1; -120,-240,-120,-20,-1; ...
%p A008297 A008297 := (n,m) -> (-1)^n*n!*binomial(n-1,m-1)/m!;
%Y A008297 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 A008297 Columns 1-6 (unsigned): A000142, A001286, A001754, A001755, A001777, 
               A001778.
%Y A008297 Cf. A001263. A111596 (differently signed triangle with extra column m=0 
               and row n=0).
%Y A008297 Sequence in context: A091599 A066667 A105278 this_sequence A048999 A090582 
               A079641
%Y A008297 Adjacent sequences: A008294 A008295 A008296 this_sequence A008298 A008299 
               A008300
%K A008297 sign,tabl,nice
%O A008297 1,2
%A A008297 N. J. A. Sloane (njas(AT)research.att.com).
%E A008297 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 03 2001

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