I am interested in the following general double sums, for integers $a\geq 1$ and $b\geq 2$,


$$Z(a,b) = \sum\_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^a} \frac{2\ell+3}{(\binom{k+2}{2}+\binom{\ell+2}{2})^b},$$


which are converging very slowly. For these sums, there is also an alternative expression as an iterated integral in dimension $a+b$, similar to multiple zeta values.


I would like more particularly to find $Z(2,2)$ and $Z(1,3)$ with precision as large as possible. Using the naive summation, I could only obtain 4 decimal digits, namely $Z(2,2) \simeq 4.7058$ and $Z(1,3) \simeq 1.6470$.


It is known that $2 Z(2,2) + 4 Z(1,3) = 16$.



> 
> What would be a smart way to accelerate the convergence, in general and in the special case using maybe the previous formula ?
> 
> 
> 



Accelerating convergence for some double sums
This is easy to do with [PARI/GP](https://pari.math.u-bordeaux.fr/).
Here is my code



```
p(n) = binomial(n+2,2);
Y(k,b) = sumnum(l=0, (2*l+3)/(p(k)+p(l))^b,sumtable);
Z(a,b) = sumnum(k=0, (2*k+3)/p(k)^a*Y(k,b));
default(realprecision,57);
sumtable = sumnuminit();
print(2*Z(2,2)+4*Z(1,3))
/* 16.0000000000000000000000000000000000000000000000000000000 */

```

It takes 1.361 CPU seconds for 57 decimal places. The
[documentation](https://pari.math.u-bordeaux.fr/doc.html)
has some information about the methods being used and there
are several summation functions other than `sumnum`. For
example, I originally used `sumpos` but `sumnum` is faster.
Thanks to Henri Cohen for his comment to use `sumnuminit`
to speed up the calculation of `Y(k,b)`. Thanks to Jorge
Zuniga for his comment that replacing `sumnum` with `summonien`
is much faster. And especially thanks to the developers of
PARI/GP who provided these fast numerical summation functions.


Some details are in the arXiv paper
[Gaussian Summation: An Exponentially Converging Summation Scheme](https://arxiv.org/abs/math/0611057)
by Hartmut Monien. (published in 2010 in
[Mathematics of Computation](https://doi.org/10.1090/S0025-5718-09-02289-3)).


