1,2

|a(n,k)| = number of partitions of {1,..,n} into k lists, where a list means an ordered subset.

T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.

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

D. E. Knuth, Convolution polynomials, The Mathematica J., 2.1 (1992) 67-78.

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+}.

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.

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

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

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

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(AT)physik.uni-karlsruhe.de

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

E.g.f. for k-th column (unsigned): x^k/(1-x)^k/k!. - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 03 2002

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

|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).

-1; 2,1; -6,-6,-1; 24,36,12,1; -120,-240,-120,-20,-1; ...

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

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).

Sequence in context: A091599 A066667 A105278 this_sequence A048999 A090582 A079641

Adjacent sequences: A008294 A008295 A008296 this_sequence A008298 A008299 A008300

sign,tabl,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 03 2001