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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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6
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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
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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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T. D. Noe, Rows n=0..50 of triangle, flattened
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].
Index entries for sequences related to Laguerre polynomials
Eric Weisstein's World of Mathematics, Rook Polynomial
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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 (Deutsch)
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MATHEMATICA
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Flatten[ Table[ Reverse[ CoefficientList[n!*LaguerreL[n, x], x]], {n, 0, 9}]] (* From Jean-François Alcover, Nov 24 2011 *)
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CROSSREFS
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Cf. A021009, A002720.
Sequence in context: A211957 A063983 A144084 * A193607 A075397 A049429
Adjacent sequences: A021007 A021008 A021009 * A021011 A021012 A021013
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KEYWORD
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sign,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers
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STATUS
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approved
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