So basically what i'm trying to find is:
-Start with $\sum_{i=1}^j 1$, which obviously is $j$
-Do the summation of the answer, which is: $\sum_{j=1}^k j$, which is $k*(k+1)/2$
-Next step is the summation of all of this: $\sum_{k=1}^l k*(k+1)/2$, what is equal to $l*(l+1)*(l+6)/12$
So, my question is: is there a formula or an easy way to know the result of this after n steps?
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You have a typo, should be $\sum_{k=1}^\ell \frac{k(k+1)}{2} = \frac{\ell(\ell+1)(\ell+\color{red}{2})}{\color{red}{6}}$. In general, the $n$-th iterated sum is $$ \sum_{k_n=1}^\ell\sum_{k_{n-1}=1}^{k_n}\cdots\sum_{k_1=1}^{k_2} 1 = \binom{\ell+n-1}{n}$$ – achille hui Feb 27 '19 at 22:20