T6:=orthpoly::chebyshev1(6, t);

poly(32*t^6 - 48*t^4 + 18*t^2 - 1, [t])

f:=x->arccos(x);

x -> arccos(x)

c:=k->2/PI*int(f(cos(x))*cos(k*x),x=0..PI);

k -> (2/PI)*int(f(cos(x))*cos(k*x), x = 0..PI)

c(0);

PI

c(1);

-4/PI

c(2);

0

c(3);

-4/(9*PI)

c(n) assuming(n, Type::NonNegInt);

-(2*(2*sin((PI*n)/2)^2 - PI*n*sin(PI*n)))/(PI*n^2)

cc:=simplify(%);

((-1)^(3*n) + (-1)^n - 2)/(PI*n^2)

simplify(cc) assuming(n, Type::Odd);

-4/(PI*n^2)

simplify(cc) assuming(n, Type::Even);

0