Recurrence for the Inner Sum

We next compute the right-hand side, beginning with the Franel numbers given by

.

We first obtain a system to describe the summand:

>

>

>

> sys:=collect(map(numer,{"","}),h);

Summing, we get a recurrence for the Franel numbers .

> sys:=sum_of_sys(sys,k=-infinity..infinity,natural_boundaries);