A178111 Number triangle T(n,k)=(-1)^((n-k)/2)*C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, -1, 0, 3, 0, -3, 0, 1, 0, -1, 0, 3, 0, -3, 0, 1, 1, 0, -4, 0, 6, 0, -4, 0, 1, 0, 1, 0, -4, 0, 6, 0, -4, 0, 1, -1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1, 0, -1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1, 1, 0, -6, 0, 15, 0, -20, 0, 15, 0, -6, 0, 1

Offset: 0

Author

Paul Barry, May 20 2010

Keywords

• easy
• sign
• tabl

Comments

Coefficient array of orthogonal polynomials P(n,x)=xP(n-1,x)-((1+(-1)^n)/2)*P(n-2,x), P(0,x)=1,P(1,x)=x.

Inverse is A178112.

Examples

			Triangle begins
1,
0, 1,
-1, 0, 1,
0, -1, 0, 1,
1, 0, -2, 0, 1,
0, 1, 0, -2, 0, 1,
-1, 0, 3, 0, -3, 0, 1,
0, -1, 0, 3, 0, -3, 0, 1,
1, 0, -4, 0, 6, 0, -4, 0, 1,
0, 1, 0, -4, 0, 6, 0, -4, 0, 1,
-1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1
Production matrix is
0, 1,
-1, 0, 1,
0, 0, 0, 1,
0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Production matrix of inverse is
0, 1,
1, 0, 1,
0, 0, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
		

Programs

Maple

P := (n,x) -> `if`(n < 2, x^n, x*P(n-1,x) - ((1+(-1)^n)/2)*P(n-2,x)):
ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(P(n,x), x),n=0..12)]);
# Peter Luschny, Aug 10 2019