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Some deleted answers! Some of them are accepted and the other are correct. I just would like to keep them at hand.

(1)- (2)- (3)- (4)- (5).

Is there a closed form for $\sum_{j=1}^{n} j^2\log{j}$?

$$ -\zeta' \left( -2 \right) +\zeta_1' \left( -2,n+1 \right). $$

A more general case

$$ \sum_{k=1}^{n} j^{s}\ln(j)^{m}= \lim_{\alpha \to s } \zeta^{(m)}_{\alpha}( - \alpha ) + \zeta^{(m)}_{\alpha} \left( -\alpha,n+1 \right). $$


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