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

\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 .