0,4

In absolute values, this sequence also gives the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals (j-1)^2 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006

A partial permutation on a set X is a bijection between two subsets of X. |T(n,n-k)| equals the numbers of partial permutations of an n-set having domain cardinality equal to k. Let E denote the operator D*x*D, where D is the derivative operator d/dx. Then E^n = sum {k = 0..n} |T(n,k)|*x^k*D^(n+k). [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

T. D. Noe, Rows n=0..50 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Index entries for sequences related to Laguerre polynomials

Eric Weisstein's World of Mathematics, Laguerre Polynomial

a(n, m)= ((-1)^m)*n!*binomial(n, m)/m! = ((-1)^m)*((n!/m!)^2)/(n-m)! if n >= m else 0. E.g.f. for m-th column: (-x/(1-x))^m /((1-x)*m!), m >= 0.

Representation (of unsigned a(n, m)) as special values of Gauss hypergeometric function 2F1, in Maple notation: n!*(-1)^m*hypergeom([ -m, n+1 ], [ 1 ], 1)/m!, from Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 02 2003

Sum_{m>=0} (-1)^m*a(n, m) = A002720(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 10 2004

E.g.f.: (1/(1-x))*exp(x*y/(x-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 07 2005

Sum_{n>=0, m>=0} a(n, m)*x^n/n!^2*y^m = exp(x)*BesselJ(0, 2*sqrt(x*y)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 07 2005

Matrix square yields the identity matrix: L^2 = I. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 22 2008]

1; 1,-1; 2,-4,1; 6,-18,9,-1; 24,-96,72,-16,1; ...

Cf. A021010, A062137-A062140, A066667. Row sums give A009940. Column sequences (unsigned): A000142, A001563, A001809-A001812 for m=0..5.

Cf. A025166, A025167.

Sequence in context: A167546 A011369 A110877 this_sequence A137478 A089087 A142146

Adjacent sequences: A021006 A021007 A021008 this_sequence A021010 A021011 A021012

sign,tabl,easy,nice

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

More terms from James A. Sellers (sellersj(AT)math.psu.edu)