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A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x). 13
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-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
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
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
KEYWORD
sign,easy,tabl
AUTHOR
Wolfdieter Lang, Jun 19 2001
STATUS
approved

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Last modified July 25 01:41 EDT 2024. Contains 374585 sequences. (Running on oeis4.)