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Question Number 124718 by Dwaipayan Shikari last updated on 05/Dec/20

$$\underset{n=1}{\sum ^n}{n}^{2}log(n+1)\left(Inexplicitform\right)$$

Commented by Dwaipayan Shikari last updated on 05/Dec/20

$$In{Ramanujan}^{\prime }sNotebook$$$$1^{2}log\left(1\right)+2^{2}log\left(2\right)+3^{2}log\left(3\right)+4^{2}log\left(4\right)+...+n^{2}log\left(n\right)$$$$=\frac{n(n+1)(2n+1)}{6}log\left(n\right)-\frac{n^3}{9}+\frac{\zeta \left(3\right)}{4{\pi }^2}+\frac{n}{12}-\frac{1}{360n}+...+C$$$$Buti{haven}^{\prime }tfoundanyproof$$

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