A199916 - OEIS

%I #11 Jan 01 2013 07:08:09

%S 1,0,1,-1,0,1,0,-5,0,1,1,0,-14,0,1,0,21,0,-30,0,1,-1,0,147,0,-55,0,1,

%T 0,-85,0,627,0,-91,0,1,1,0,-1408,0,2002,0,-140,0,1,0,341,0,-11440,0,

%U 5278,0,-204,0,1,-1,0,13013,0,-61490,0,12138,0,-285,0,1,0

%N Triangle T(n,k) = coefficient of x^n in expansion of ((2-2*cos(x))/x)^k = sum(n>=k, T(n,k) * x^n * (2*k)!/(n+k)!).

%C Triangle T(n,k)*(2*k)!/(n+k)!)=

%C 1. Riordan Array (1,(2-2*cos(x))/x) without first column.

%C 2. Riordan Array ((2-2*cos(x))/x^2,(2-2*cos(x))/x) numbering triangle (0,0).

%F T(n,k):=((-1)^k*2^k*((-1)^(n+k)+1)*sum(j=1..k, ((sum(i=0..(j-1)/2, (j-2*i)^(n+k)*binomial(j,i))) *binomial(k,j)*(-1)^((n+k)/2+k-j))/2^j))/(2*k)!

%e 1

%e 0, 1

%e -1, 0, 1

%e 0, -5, 0, 1

%e 1, 0, -14, 0, 1

%e 0, 21, 0, -30, 0, 1

%e -1, 0, 147, 0, -55, 0, 1

%o (Maxima)

%o T(n,k):=((-1)^k*2^k*((-1)^(n+k)+1)*sum(((sum((j-2*i)^(n+k)*binomial(j,i),i,0,(j-1)/2))*binomial(k,j)*(-1)^((n+k)/2+k-j))/2^j,j,1,k))/(2*k)!

%K sign,tabl

%O 1,8

%A _Vladimir Kruchinin_, Nov 11 2011