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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.

%I A178111 #5 Aug 10 2019 04:31:00
%S A178111 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,
%T A178111 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,
%U A178111 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
%N A178111 Number triangle T(n,k)=(-1)^((n-k)/2)*C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.
%C A178111 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.
%C A178111 Inverse is A178112.
%e A178111 Triangle begins
%e A178111 1,
%e A178111 0, 1,
%e A178111 -1, 0, 1,
%e A178111 0, -1, 0, 1,
%e A178111 1, 0, -2, 0, 1,
%e A178111 0, 1, 0, -2, 0, 1,
%e A178111 -1, 0, 3, 0, -3, 0, 1,
%e A178111 0, -1, 0, 3, 0, -3, 0, 1,
%e A178111 1, 0, -4, 0, 6, 0, -4, 0, 1,
%e A178111 0, 1, 0, -4, 0, 6, 0, -4, 0, 1,
%e A178111 -1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1
%e A178111 Production matrix is
%e A178111 0, 1,
%e A178111 -1, 0, 1,
%e A178111 0, 0, 0, 1,
%e A178111 0, 0, -1, 0, 1,
%e A178111 0, 0, 0, 0, 0, 1,
%e A178111 0, 0, 0, 0, -1, 0, 1,
%e A178111 0, 0, 0, 0, 0, 0, 0, 1,
%e A178111 0, 0, 0, 0, 0, 0, -1, 0, 1,
%e A178111 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
%e A178111 Production matrix of inverse is
%e A178111 0, 1,
%e A178111 1, 0, 1,
%e A178111 0, 0, 0, 1,
%e A178111 0, 0, 1, 0, 1,
%e A178111 0, 0, 0, 0, 0, 1,
%e A178111 0, 0, 0, 0, 1, 0, 1,
%e A178111 0, 0, 0, 0, 0, 0, 0, 1,
%e A178111 0, 0, 0, 0, 0, 0, 1, 0, 1,
%e A178111 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
%p A178111 P := (n,x) -> `if`(n < 2, x^n, x*P(n-1,x) - ((1+(-1)^n)/2)*P(n-2,x)):
%p A178111 ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(P(n,x), x),n=0..12)]);
%p A178111 # _Peter Luschny_, Aug 10 2019
%K A178111 easy,sign,tabl
%O A178111 0,13
%A A178111 _Paul Barry_, May 20 2010