# A geometric-like sum with N variables

I want to compute the following sum

$F_N=\sum_{1\le n_1<n_2\cdots<n_N\le L} a_1^{n_1} a_2^{n_2}\cdots a_N^{n_N}$

for rather large $L$. Is there a general formula for this kind of sum ?

• It's doable, but is pretty unpleasant in general. I doubt there's a reasonable closed form. – Chappers Jun 11 '15 at 15:42