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

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, December 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.yu), 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.yu), Apr 07 2005

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: A105357 A011369 A110877 this_sequence A137478 A089087 A119303

Adjacent sequences: A021006 A021007 A021008 this_sequence A021010 A021011 A021012

sign,tabl,easy,nice

njas

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