**Recurrence for the Left-Hand Side**

We first prove that the Apéry numbers, as defined by the left-hand side

,

satisfy the announced recurrence.

The summand

`> `

satisfies both following equations:

`> `
**h(n+1,k)/h(n,k)=factor(normal(subs(n=n+1,f)/f,expanded));**

`> `
**h(n,k+1)/h(n,k)=factor(normal(subs(k=k+1,f)/f,expanded));**

This yields the following system

`> `
**sys:=collect(map(numer,map(eq->op(1,eq)-op(2,eq),{"","})),h);**

where each element
in the set denotes the equation
. The definite summation over
in
is performed by the following call to
__Mgfun[sum_of_sys]__
:

`> `
**sum_of_sys(sys,k=-infinity..infinity,takayama_algo);**

`> `
**rec[left]:=op(collect(",h,factor));**

This is the announced recurrence in disguise.