A128411 Coefficient array for orthogonal polynomials defined by C(2n,n).

1, -2, 1, 4, -8, 2, -8, 36, -24, 4, 16, -128, 160, -64, 8, -32, 400, -800, 560, -160, 16, 64, -1152, 3360, -3584, 1728, -384, 32, -128, 3136, -12544, 18816, -13440, 4928, -896, 64, 256, -8192, 43008, -86016, 84480, -45056, 13312

Offset: 0

Author

Paul Barry, Mar 02 2007

Keywords

• easy
• sign
• tabl

Comments

Define {p(n,x)} to be the family of orthogonal polynomials on [0,4] for the weight function (1/pi)*1/sqrt(x(4-x)) which defines C(2n,n). We have p(n,x)=(2x-4)*p(n-1,x)-4*p(n-2,x), with p(0,x)=1, p(1,x)=-2+x. A scaled version of this triangle is given by A128412.

Examples

			Triangle begins
1,
-2, 1,
4, -8, 2,
-8, 36, -24, 4,
16, -128, 160, -64, 8,
-32, 400, -800, 560, -160, 16,
64, -1152, 3360, -3584, 1728, -384, 32
		

Formula

Column k has g.f. if(k=0,1/(1+2x),(1-2x)*((2^(k-1)+0^k/2)*x^k/(1+2x)^(2k+1))).

T(n,k)=(C(n+k,n-k)(-1)^(n-k)-C(n+k-1,n-k-1)(-1)^(n-k-1))*(2^(n-1)+0^n/2); T(n,k)=A110162(n,k)*(2^(n-1)+0^n/2); - Paul Barry, Mar 22 2007