Question

how does one determine the generating function for the series of squares {0^2,1^2,2^2,3^2,.....,n^2}

Answer 1

Start from \(\frac{x^{n+1}-1}{x-1}=1+x+x^2+\cdots+x^n\) and do a series of differentiations and multiplications to transform RHS to \(1^2x+2^2x^2+\cdots+n^2x^n\).