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A021010
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Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).
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8
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1, -1, 1, 1, -4, 2, -1, 9, -18, 6, 1, -16, 72, -96, 24, -1, 25, -200, 600, -600, 120, 1, -36, 450, -2400, 5400, -4320, 720, -1, 49, -882, 7350, -29400, 52920, -35280, 5040, 1, -64, 1568, -18816, 117600, -376320, 564480, -322560, 40320, -1, 81, -2592, 42336, -381024, 1905120, -5080320, 6531840, -3265920, 362880
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OFFSET
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0,5
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COMMENTS
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abs(T(n,k)) = k!*binomial(n,k)^2 = number of k-matchings of the complete bipartite graph K_{n,n}. Example: abs(T(2,2))=2 because in the bipartite graph K_{2,2} with vertex sets {A,B},{A',B'} we have the 2-matchings {AA',BB'} and {AB',BA'}. Row sums of the absolute values yield A002720. - Emeric Deutsch, Dec 25 2004
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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T(n, k) = (-1)^(n-k)*k!*binomial(n, k)^2. - Emeric Deutsch, Dec 25 2004
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EXAMPLE
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1;
-1, 1;
1, -4, 2;
-1, 9, -18, 6;
1, -16, 72, -96, 24;
...
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MAPLE
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T:=(n, k)->(-1)^(n-k)*k!*binomial(n, k)^2: for n from 0 to 9 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form # Emeric Deutsch, Dec 25 2004
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MATHEMATICA
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Flatten[ Table[ Reverse[ CoefficientList[n!*LaguerreL[n, x], x]], {n, 0, 9}]] (* Jean-François Alcover, Nov 24 2011 *)
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PROG
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(PARI)
LaguerreL(n, v='x) = {
my(x='x+O('x^(n+1)), t='t);
subst(polcoeff(exp(-x*t/(1-x))/(1-x), n), 't, v);
};
(PARI) row(n) = Vec(n!*pollaguerre(n)); \\ Michel Marcus, Feb 06 2021
(Magma) [[(-1)^(n-k)*Factorial(k)*Binomial(n, k)^2: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018
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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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