Search: id:A021009
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%I A021009
%S A021009 1,1,1,2,4,1,6,18,9,1,24,96,72,16,1,120,600,600,200,25,1,720,
%T A021009 4320,5400,2400,450,36,1,5040,35280,52920,29400,7350,882,49,1,
%U A021009 40320,322560,564480,376320,117600,18816,1568,64,1,362880,3265920
%V A021009 1,1,-1,2,-4,1,6,-18,9,-1,24,-96,72,-16,1,120,-600,600,-200,25,-1,720,
%W A021009 -4320,5400,-2400,450,-36,1,5040,-35280,52920,-29400,7350,-882,49,-1,
%X A021009 40320,-322560,564480,-376320,117600,-18816,1568,-64,1,362880,-3265920
%N A021009 Triangle of coefficients of Laguerre polynomials L_n(x).
%C A021009 In absolute values, this sequence also gives the lower triangular readout 
               of the exponential of a matrix whose entry {j+1,j} equals (j-1)^2 
               (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), 
               May 26 2006
%C A021009 A partial permutation on a set X is a bijection between two subsets of 
               X. |T(n,n-k)| equals the numbers of partial permutations of an n-set 
               having domain cardinality equal to k. Let E denote the operator D*x*D, 
               where D is the derivative operator d/dx. Then E^n = sum {k = 0..n} 
               |T(n,k)|*x^k*D^(n+k). [From Peter Bala (pbala(AT)toucansurf.com), 
               Oct 28 2008]
%D A021009 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 A021009 T. D. Noe, <a href="b021009.txt">Rows n=0..50 of triangle, flattened</
               a>
%H A021009 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A021009 <a href="Sindx_La.html#Laguerre">Index entries for sequences related 
               to Laguerre polynomials</a>
%H A021009 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LaguerrePolynomial.html">Laguerre Polynomial</a>
%F A021009 a(n, m)= ((-1)^m)*n!*binomial(n, m)/m! = ((-1)^m)*((n!/m!)^2)/(n-m)! 
               if n >= m else 0. E.g.f. for m-th column: (-x/(1-x))^m /((1-x)*m!), 
               m >= 0.
%F A021009 Representation (of unsigned a(n, m)) as special values of Gauss hypergeometric 
               function 2F1, in Maple notation: n!*(-1)^m*hypergeom([ -m, n+1 ], 
               [ 1 ], 1)/m!, from Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 
               02 2003
%F A021009 Sum_{m>=0} (-1)^m*a(n, m) = A002720(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 10 2004
%F A021009 E.g.f.: (1/(1-x))*exp(x*y/(x-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 07 2005
%F A021009 Sum_{n>=0, m>=0} a(n, m)*x^n/n!^2*y^m = exp(x)*BesselJ(0, 2*sqrt(x*y)). 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 07 2005
%F A021009 Matrix square yields the identity matrix: L^2 = I. [From Paul D. Hanna 
               (pauldhanna(AT)juno.com), Nov 22 2008]
%e A021009 1; 1,-1; 2,-4,1; 6,-18,9,-1; 24,-96,72,-16,1; ...
%Y A021009 Cf. A021010, A062137-A062140, A066667. Row sums give A009940. Column 
               sequences (unsigned): A000142, A001563, A001809-A001812 for m=0..5.
%Y A021009 Cf. A025166, A025167.
%Y A021009 Sequence in context: A167546 A011369 A110877 this_sequence A137478 A089087 
               A142146
%Y A021009 Adjacent sequences: A021006 A021007 A021008 this_sequence A021010 A021011 
               A021012
%K A021009 sign,tabl,easy,nice
%O A021009 0,4
%A A021009 N. J. A. Sloane (njas(AT)research.att.com).
%E A021009 More terms from James A. Sellers (sellersj(AT)math.psu.edu)

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