convolutions and sum of sums

Let S_k be the sequence of simple sums, that is S₁=1, S₂ =1+2, S_k =1+2+...+k then

S₁ + S₂ + ... + S_k = k.1+(k-1).2+(k-2).3+ ... + 2.(k-1)+1.k = the convolution of the sequence {1, ...,k}.

In your spare time you can show that sum of sums equals (n)(n+1)(n+2)/(1.2.3) and sum of sum of sums equals n(n+1)(n+2)(n+3)/(1.2.3.4) and so on.

1.2.3 = 6.
1.2.3.4 = 24.