OFFSET
0,2
COMMENTS
The row polynomials s(n,x) := n!*L(n,3,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^4. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} |A008297(n,m)|*(-x)^m, 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=1 (p-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
The unsigned version of this triangle is the triangle of unsigned 2-Lah numbers A143497. - Peter Bala, Aug 25 2008
This matrix (unsigned) is embedded in that for n!*L(n,-3,-x). Introduce 0,0,0 to each unsigned row and then add 1,-2,1,4,2,1 to the beginning of the array as the first three rows to generate n!*L(n,-3,-x). - Tom Copeland, Apr 21 2014
From Wolfdieter Lang, Jul 07 2014: (Start)
The standard Rodrigues formula for the monic generalized Laguerre polynomials (also called Laguerre-Sonin polynomials) is Lm(n,alpha,x) := (-1)^n*n!*L(n,3,x) is x^(-alpha)*exp(x)*(d/dx)^n(exp(-x)*x^(n+alpha)).
Another Rodrigues type formula is Lm(n,alpha,x) = exp(x)*x^(-alpha+n+1)*(-x^2*d/dx)^n*(exp(-x)*x^(alpha+1)). This is derivable from the differential - difference relation of Lm(n,alpha,x) which yields first the creation operator formula -(x*d/dx + (-x + alpha + n + 1))*Lm(n,alpha,x) = Lm(n+1,alpha,x) or in the variable q = log(x) the operator -(d/dq + alpha + n + 1 - exp(q)).
The identity (due to Christoph Mayer) (d/dq - (d/dq)W(q))*f(q) = exp(W(q)*d/dq(exp(-W(q)*f(q)) is then iterated with W(q) = W(alpha,n,q) = exp(q) - (alpha + n + 1)*q and a further change of variables leads then to the given result. (End)
REFERENCES
A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.
LINKS
Indranil Ghosh, Rows 0..125, flattened
Wolfdieter Lang, First eleven rows of the triangle.
FORMULA
a(n, m) = ((-1)^m)*n!*binomial(n+3, n-m)/m!.
E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^4), m >= 0.
EXAMPLE
The triangle a(n,m) begins:
n\m 0 1 2 3 4 5 ...
0: 1
1: 4 -1
2: 20 -10 1
3: 120 -90 18 -1
4: 840 -840 252 -28 1
5: 6720 -8400 3360 -560 40 -1
... Formatted by Wolfdieter Lang, Jul 07 2014
For more rows see the link.
n = 2: 2!*L(2,3,x) = 20 - 10*x + x^2.
MATHEMATICA
Flatten[Table[((-1)^m)*n!*Binomial[n+3, n-m]/m!, {n, 0, 9}, {m, 0, n}]] (* Indranil Ghosh, Feb 23 2017 *)
PROG
(PARI) row(n) = Vecrev(n!*pollaguerre(n, 3)); \\ Michel Marcus, Feb 06 2021
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Jun 19 2001
STATUS
approved