Norms Related to Binomial Series

We investigate norms related to the vectors h(n) = (h(n)(k))(k is an element of N) in R-N (or C-N) (n is an element of N) where h(n)(k) = n!/k(k + 1) ... (k + n) = 1/ k((n+k)(n)). We estimate, and in a few cases even calculate, the norms of the h(n)'s as elements of the usual sequence spaces l(r). We also show that for 'almost all' p, q the matrix H with entries h(n)(k) defines a bounded linear operator from l(p) into l(q), with rather strong compactness properties.