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A178111
Number triangle T(n,k)=(-1)^((n-k)/2)*C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.
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.
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
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, May 20 2010
STATUS
approved