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A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x). 10
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 (list; table; graph; refs; listen; history; text; internal format)
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-restricted Lah numbers A143498. [From 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

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.

EXAMPLE

{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

(Python)

import math

f=math.factorial

def C(n, r):return f(n)/f(r)/f(n-r)

i=0

for n in range(0, 126):

....for m in range(0, n+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-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, Aug 25 2008]

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

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Last modified February 26 17:19 EST 2020. Contains 332293 sequences. (Running on oeis4.)