  We define two notions of partial sums of a Riordan array, corresponding
respectively to the partial sums of the rows and the partial sums of the
columns of the Riordan array in question. We characterize the matrices that
arise from these operations. On the one hand, we obtain a new Riordan array,
while on the other hand, we obtain a rectangular array which has an inverse
that is a lower Hessenberg matrix. We examine the structure of these Hessenberg
matrices. We end with a generalization linked to the Fibonacci numbers and
phyllotaxis.
