0,2

The row polynomials s(n,x) := n!*L(n,5,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^6. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials sum(|A008297(n,m)|*(-x)^m, m=1..n), n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

These polynomials appear in the radial part of the l=2 (d-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.

For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148-A062152. Row sums (signed) give A062191; row sums (unsigned) give A062192.

The unsigned version of this triangle is the triangle of unsigned 3-restricted Lah numbers A143498. [From Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008]

A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.

Index entries for sequences related to Laguerre polynomials

a(n, m)=((-1)^m)*n!*binomial(n+5, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^6), m >= 0.

{1}; {6, -1}; {42, -14, 1}; {336, -168, 24, -1}; ...; 2!*L(2, 5, x) = 42-14*x+x^2.

Cf. A021009, A062137-A062140, A066667.

For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148-A062152. Row sums (signed) give A062191, row sums (unsigned) give A062192.

A143498. [From Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008]

Sequence in context: A035529 A135893 A051338 this_sequence A143498 A144356 A049374

Adjacent sequences: A062135 A062136 A062137 this_sequence A062139 A062140 A062141

sign,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 19 2001