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%I A021010 #51 Aug 14 2022 16:06:15
%S A021010 1,-1,1,1,-4,2,-1,9,-18,6,1,-16,72,-96,24,-1,25,-200,600,-600,120,1,
%T A021010 -36,450,-2400,5400,-4320,720,-1,49,-882,7350,-29400,52920,-35280,
%U A021010 5040,1,-64,1568,-18816,117600,-376320,564480,-322560,40320,-1,81,-2592,42336,-381024,1905120,-5080320,6531840,-3265920,362880
%N A021010 Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).
%C A021010 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
%D A021010 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.
%H A021010 T. D. Noe, <a href="/A021010/b021010.txt">Rows n = 0..50 of triangle, flattened</a>
%H A021010 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A021010 J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013.
%H A021010 C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages) See page 519.
%H A021010 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookPolynomial.html">Rook Polynomial</a>
%H A021010 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A021010 T(n, k) = (-1)^(n-k)*k!*binomial(n, k)^2. - _Emeric Deutsch_, Dec 25 2004
%e A021010    1;
%e A021010   -1,   1;
%e A021010    1,  -4,   2;
%e A021010   -1,   9, -18,   6;
%e A021010    1, -16,  72, -96,  24;
%e A021010   ...
%p A021010 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
%t A021010 Flatten[ Table[ Reverse[ CoefficientList[n!*LaguerreL[n, x], x]], {n, 0, 9}]] (* _Jean-François Alcover_, Nov 24 2011 *)
%o A021010 (PARI)
%o A021010 LaguerreL(n,v='x) = {
%o A021010   my(x='x+O('x^(n+1)), t='t);
%o A021010   subst(polcoeff(exp(-x*t/(1-x))/(1-x), n), 't, v);
%o A021010 };
%o A021010 concat(apply(n->Vec(n!*LaguerreL(n)), [0..9])) \\ _Gheorghe Coserea_, Oct 26 2017
%o A021010 (PARI) row(n) = Vec(n!*pollaguerre(n)); \\ _Michel Marcus_, Feb 06 2021
%o A021010 (Magma) [[(-1)^(n-k)*Factorial(k)*Binomial(n, k)^2: k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Feb 06 2018
%Y A021010 Cf. A002720, A021009.
%Y A021010 Central terms: A295383.
%K A021010 sign,tabl,easy,nice
%O A021010 0,5
%A A021010 _N. J. A. Sloane_

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