If the coefficient of $x^r$ in the product of $(1-x+x^2-x^3+......+x^{100})(1+x+x^2+x^3+.....+x^{100})$ is denoted by $T_r$. What is the value of $(T_{99}+T_{101}+T_{103})$
I don't know how to approach the question. Do I have to evaluate all of those T's separately? Or do they simplify into something beautiful?
This polynomial is $$\frac{1+x^{101}}{1+x}\frac{1-x^{101}}{1-x} =\frac{1-x^{202}}{1-x^2}=1+x^2+x^4+\cdots+x^{200}.$$ What are $T_{99}$ etc.?