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A062139 Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x). 12
1, 3, -1, 12, -8, 1, 60, -60, 15, -1, 360, -480, 180, -24, 1, 2520, -4200, 2100, -420, 35, -1, 20160, -40320, 25200, -6720, 840, -48, 1, 181440, -423360, 317520, -105840, 17640, -1512, 63, -1, 1814400, -4838400, 4233600 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The row polynomials s(n,x) := n!*L(n,2,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^3. 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 p(n,x)= 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).

This unsigned matrix is embedded in the matrix for n!*L(n,-2,-x). Introduce 0,0 to each unsigned row and then add 1,-1,1 to the array as the first two rows to generate n!*L(n,-2,-x). - Tom Copeland, Apr 20 2014

The unsigned n-th row reverse polynomial equals the numerator polynomial of the finite continued fraction 1 - x/(1 + (n+1)*x/(1 + n*x/(1 + n*x/(1 + ... + 2*x/(1 + 2*x/(1 + x/(1 + x/(1)))))))). Cf. A089231. The denominator polynomial of the continued fraction is the (n+1)-th row polynomial of A144084. An example is given below. - Peter Bala, Oct 06 2019

LINKS

Indranil Ghosh, Rows 0..125, flattened

Index entries for sequences related to Laguerre polynomials

FORMULA

a(n, m) = ((-1)^m)*n!*binomial(n+2, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^3), m >= 0.

n!*L(n,2,x) = (n+2)!*hypergeom([-n],[3],x)/2. - Peter Luschny, Apr 08 2015

EXAMPLE

Triangle begins:

     1;

     3,    -1;

    12,    -8,    1;

    60,   -60,   15,   -1;

   360,  -480,  180,  -24,  1;

  2520, -4200, 2100, -420, 35, -1;

  ...

2!*L(2,2,x) = 12 - 8*x + x^2.

Unsigned row 3 polynomial in reverse form as the numerator of a continued fraction: 1 - x/(1 + 4*x/(1 + 3*x/(1 + 3*x/(1 + 2*x/(1 + 2*x/(1 + x/(1 + x))))))) = (60*x^3 + 60*x^2 + 15*x + 1)/(24*x^4 + 96*x^3 + 72*x^2 + 16*x + 1). - Peter Bala, Oct 06 2019

MAPLE

with(PolynomialTools):

p := n -> (n+2)!*hypergeom([-n], [3], x)/2:

seq(CoefficientList(simplify(p(n)), x), n=0..9); # Peter Luschny, Apr 08 2015

MATHEMATICA

Flatten[Table[((-1)^m)*n!*Binomial[n+2, n-m]/m!, {n, 0, 8}, {m, 0, n}]] (* Indranil Ghosh, Feb 24 2017 *)

PROG

(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(((-1)^k)*n!*binomial(n+2, n-k)/k!, ", "); ); print(); ); } \\ Michel Marcus, May 06 2014

(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+2, n-m)/f(m)) # Indranil Ghosh, Feb 24 2017

CROSSREFS

For m=0..5 the (unsigned) columns give A001710, A005990, A005461, A062193-A062195. The row sums (signed) give A062197, the row sums (unsigned) give A052852.

Cf. A021009, A062137-A062140, A066667, A089231, A144084.

Sequence in context: A049458 A143492 A243662 * A156366 A144353 A039811

Adjacent sequences:  A062136 A062137 A062138 * A062140 A062141 A062142

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Jun 19 2001

STATUS

approved

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Last modified September 24 20:13 EDT 2020. Contains 337321 sequences. (Running on oeis4.)