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A062137 Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x). 15
1, 4, -1, 20, -10, 1, 120, -90, 18, -1, 840, -840, 252, -28, 1, 6720, -8400, 3360, -560, 40, -1, 60480, -90720, 45360, -10080, 1080, -54, 1, 604800, -1058400, 635040, -176400, 25200, -1890, 70, -1, 6652800, -13305600, 9313920 (list; table; graph; refs; listen; history; text; internal format)
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.

Index entries for sequences related to Laguerre polynomials

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 *)

CROSSREFS

For m=0..5 the (unsigned) columns give A001715, A061206, A062141-A062144. The row sums (signed) give A062146, the row sums (unsigned) give A062147.

Cf. A143497. - Peter Bala, Aug 25 2008

Cf. A062139, A105278. - Wolfdieter Lang, Jul 07 2014

Sequence in context: A135891 A049459 A143493 * A143497 A144354 A049352

Adjacent sequences:  A062134 A062135 A062136 * A062138 A062139 A062140

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Jun 19 2001

STATUS

approved

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Last modified February 25 02:54 EST 2020. Contains 332217 sequences. (Running on oeis4.)