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

0,13

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.

Links

Table of n, a(n) for n=0..90.

Example

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

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

Crossrefs

Sequence in context: A267860 A325488 A339813 * A178112 A324852 A363854

Adjacent sequences: A178108 A178109 A178110 * A178112 A178113 A178114

Keyword

easy,sign,tabl

Author

Paul Barry, May 20 2010

Status

approved