A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).

1, 6, -1, 42, -14, 1, 336, -168, 24, -1, 3024, -2016, 432, -36, 1, 30240, -25200, 7200, -900, 50, -1, 332640, -332640, 118800, -19800, 1650, -66, 1, 3991680, -4656960, 1995840, -415800, 46200, -2772, 84, -1, 51891840, -69189120

Offset

0,2

Comments

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-Lah numbers A143498. - Peter Bala, Aug 25 2008

References

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

Links

Indranil Ghosh, Rows 0..125, flattened

Index entries for sequences related to Laguerre polynomials

Formula

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

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

Example

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

Mathematica

Flatten[Table[((-1)^m)*n!*Binomial[n+5, n-m]/m!, {n, 0, 8}, {m, 0, n}]] (* Indranil Ghosh, Feb 24 2017 *)

Prog

(PARI) tabl(nn) = {for (n=0, nn, for (m=0, n, print1(((-1)^m)*n!*binomial(n+5, n-m)/m!, ", "); ); print(); ); } \\ Indranil Ghosh, Feb 24 2017
(PARI) row(n) = Vecrev(n!*pollaguerre(n, 5)); \\ Michel Marcus, Feb 06 2021
(Python)
import math
f=math.factorial
def C(n, r):return f(n)//f(r)//f(n-r)
i=-1
for n in range(26):
    for m in range(n+1):
        i += 1
        print(str(i)+" "+str(((-1)**m)*f(n)*C(n+5, n-m)//f(m))) # Indranil Ghosh, Feb 24 2017

Crossrefs

Cf. A021009, A062137, A062139, A062140, A066667.

For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148, A062149, A062150, A062151, A062152.

Row sums (signed) give A062191, row sums (unsigned) give A062192.

Cf. A143498.

Sequence in context: A035529 A135893 A051338 * A143498 A144356 A049374

Adjacent sequences: A062135 A062136 A062137 * A062139 A062140 A062141

Keyword

sign,easy,tabl

Author

Wolfdieter Lang, Jun 19 2001

Status

approved