A229789 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, -2, 5, 4, -28, 35, -8, 108, -378, 315, 16, -352, 2376, -5544, 3465, -32, 1040, -11440, 51480, -90090, 45045, 64, -2880, 46800, -343200, 1158300, -1621620, 675675, -128, 7616, -171360, 1856400, -10210200, 27567540, -32162130, 11486475, 256, -19456, 578816, -8682240, 70543200, -310390080, 698377680, -698377680, 218243025

Offset: 0

Author

R. J. Mathar, Sep 29 2013

Keywords

• sign
• tabl

Examples

			2^0*0!*L(0,3/2,x) = 1;
2^1*1!*L(1,3/2,x) = -2*x+5;
2^2*2!*L(2,3/2,x) = 4*x^2 -28*x +35 ;
2^3*3!*L(3,3/2,x) = -8*x^3+108*x^2 -378*x +315 ;
2^4*4!*L(4,3/2,x) = 16*x^4 -352*x^3 +2376*x^2 -5544*x +3465 ;
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened
• R. J. Mathar, Gauss-Laguerre and Gauss-Hermite quadrature on 64, 96 and 128 nodes, see Hermite case with m=2.

Crossrefs

Cf. A098503.

Programs

Maple

A229789 := proc(n,k)
    local p;
    p := 2^n*n!*expand(L(n,3/2,x)) ;
    coeftayl(p,x=0,n-k) ;
end proc:
seq(seq(A229789(n,k),k=0..n),n=0..10) ;

Mathematica

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 *)