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A021010 Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order). 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
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
REFERENCES
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.
LINKS
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].
J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
C. Lanczos, Applied Analysis (Annotated scans of selected pages) See page 519.
Eric Weisstein's World of Mathematics, Rook Polynomial
FORMULA
T(n, k) = (-1)^(n-k)*k!*binomial(n, k)^2. - Emeric Deutsch, Dec 25 2004
EXAMPLE
1;
-1, 1;
1, -4, 2;
-1, 9, -18, 6;
1, -16, 72, -96, 24;
...
MAPLE
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
MATHEMATICA
Flatten[ Table[ Reverse[ CoefficientList[n!*LaguerreL[n, x], x]], {n, 0, 9}]] (* Jean-François Alcover, Nov 24 2011 *)
PROG
(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);
};
concat(apply(n->Vec(n!*LaguerreL(n)), [0..9])) \\ Gheorghe Coserea, Oct 26 2017
(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
CROSSREFS
Central terms: A295383.
Sequence in context: A367178 A259985 A144084 * A342088 A193607 A358735
KEYWORD
sign,tabl,easy,nice
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)