**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);**