Arrange the t values as a row vector. You can now calculate exp(2*pi*1j*t/T) and get a row vector. Notice that the exp() part here does not include k. Remember that exp(x*k) = exp(x)^k so you can take the k into account in a later step.
Now create a column vector K=-N:N
Now take the row vector of exp() values and take it .^ K . The result will be a 2d array in which the rows correspond to different k values and the columns correspond to different times.
Now construct a column vector A of a_k values in the exact same order as K, so a_(-N) first and a_(N) last.
Multiply A by the result of the ^ operation. The result will be a 2d array in which the rows correspond to k values and the columns correspond to time.
Now total the 2d array along the rows. The result will be a vector in which the columns correspond to time.
