Convolutions show up in a lot of places. Multiplying polynomials is a convolution. This provides an interesting technique to create all the partial sums of a se

Convolutions show up in a lot of places. Multiplying polynomials is a convolution. This provides an interesting technique to create all the partial sums of a sequence. If you have a generating function for the sequence, divide it by (1-x). You'll get a new generating function where the nth coefficient is the sum of the first n coefficients of the original function. Remember 1/(1-x)=1+x×x^2+... ?