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A062139
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Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x).
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16
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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
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OFFSET
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0,2
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COMMENTS
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The row polynomials s(n,x) := n!*L(n,2,x) = Sum_{m=0..n} a(n,m)*x^m 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_{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).
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
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LINKS
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FORMULA
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T(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
T(n, k) = (n+k+2) * T(n-1, k) - T(n-1, k-1) with initial values T(0, 0) = 1 and T(i, j) = 0 if j < 0 or j > i.
T = T^(-1), i.e., T is matrix inverse of T. (End)
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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Flatten[Table[((-1)^m)*n!*Binomial[n+2, n-m]/m!, {n, 0, 8}, {m, 0, n}]] (* Indranil Ghosh, Feb 24 2017 *)
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PROG
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(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
(PARI) row(n) = Vecrev(n!*pollaguerre(n, 2)); \\ Michel Marcus, Feb 06 2021
(Python)
import math
f=math.factorial
def C(n, r):return f(n)//f(r)//f(n-r)
i=0
for n in range(16):
for m in range(n+1):
i += 1
print(i, ((-1)**m)*f(n)*C(n+2, n-m)//f(m)) # Indranil Ghosh, Feb 24 2017
(Python)
from functools import cache
@cache
def T(n, k):
if k < 0 or k > n: return 0
if k == n: return (-1)**n
return (n + k + 2) * T(n-1, k) - T(n-1, k-1)
for n in range(7): print([T(n, k) for k in range(n + 1)])
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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