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A062138
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Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).
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13
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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
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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,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.
The unsigned version of this triangle is the triangle of unsigned 3-Lah numbers A143498. - Peter Bala, Aug 25 2008
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REFERENCES
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A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.
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LINKS
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FORMULA
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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.
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EXAMPLE
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Triangle begins:
{1};
{6, -1};
{42, -14, 1};
{336, -168, 24, -1};
...
2!*L(2, 5, x) = 42-14*x+x^2.
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MATHEMATICA
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Flatten[Table[((-1)^m)*n!*Binomial[n+5, 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 (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
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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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