Product of sums which equal to sum of product We can be sure that
$$\left(\sum\limits_{k=0}^{n}\frac{1}{k+1}\right)\left(\sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{k+1}\right)= \sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{(k+1)^2}$$
Is there any similar identities or some types of generalization to find them?

Let $e_k=1$. You want sequences $a_k,\,b_k$ with $\sum_k a_k \sum_k b_k =\sum_k a_k b_k$, or in terms of inner products $a\cdot e \, e\cdot b = a\cdot b$. This equation would be easy to satisfy for $3$-dimensional  vectors: choose your favourite vector $c$ and take $a=c\times e,\, b=a\times c$ so $a\cdot e =a\cdot b =0$. And now you can stitch together infinitely many triples for the infinite-sequence problem, scaling the triples as you go so the resulting sums are convergent. So there are clearly a lot of solutions.
