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Triangle T(n,k): the coefficient [x^(n-k)] of the polynomial 2^n*n!*L(n,3/2,x), where L is the generalized Laguerre Polynomial in the Abramowitz-Stegun normalization.
1

%I #13 May 15 2018 08:35:42

%S 1,-2,5,4,-28,35,-8,108,-378,315,16,-352,2376,-5544,3465,-32,1040,

%T -11440,51480,-90090,45045,64,-2880,46800,-343200,1158300,-1621620,

%U 675675,-128,7616,-171360,1856400,-10210200,27567540,-32162130,11486475,256,-19456,578816,-8682240,70543200,-310390080,698377680,-698377680,218243025

%N Triangle T(n,k): the coefficient [x^(n-k)] of the polynomial 2^n*n!*L(n,3/2,x), where L is the generalized Laguerre Polynomial in the Abramowitz-Stegun normalization.

%H G. C. Greubel, <a href="/A229789/b229789.txt">Rows n=0..100 of triangle, flattened</a>

%H R. J. Mathar, <a href="http://vixra.org/abs/1303.0013">Gauss-Laguerre and Gauss-Hermite quadrature on 64, 96 and 128 nodes</a>, see Hermite case with m=2.

%e 2^0*0!*L(0,3/2,x) = 1;

%e 2^1*1!*L(1,3/2,x) = -2*x+5;

%e 2^2*2!*L(2,3/2,x) = 4*x^2 -28*x +35 ;

%e 2^3*3!*L(3,3/2,x) = -8*x^3+108*x^2 -378*x +315 ;

%e 2^4*4!*L(4,3/2,x) = 16*x^4 -352*x^3 +2376*x^2 -5544*x +3465 ;

%p A229789 := proc(n,k)

%p local p;

%p p := 2^n*n!*expand(L(n,3/2,x)) ;

%p coeftayl(p,x=0,n-k) ;

%p end proc:

%p seq(seq(A229789(n,k),k=0..n),n=0..10) ;

%t t[n_, k_] := Coefficient[2^n*n!*LaguerreL[n, 3/2, x], x, n-k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 09 2014 *)

%Y Cf. A098503.

%K sign,tabl

%O 0,2

%A _R. J. Mathar_, Sep 29 2013