how can we solve for $\displaystyle a_{k}$ in the linear system :

$\displaystyle \sum^{n}_{k=0}(-1)^{k}a_{k}[(k+1)^0 (k+1)^1 (k+1)^2 ..... (k+1)^n]^{T}=[1 0 0 .... 0]^T$

of course i'am talking about the general case , not for specified values of n .