Charcoal, 12 bytes

ＩΣＥ⊕Ｎ⁻Ｘι³Ｘι²

Try it online! Link is to verbose version of code. Explanation: \$ ( \sum x )^2 = \sum x^3 \$

    Ｎ           Input number
   ⊕            Incremented
  Ｅ             Map over implicit range i.e. 0 .. n
       ι  ι     Current value
        ³       Literal 3
           ²    Literal 2
      Ｘ  Ｘ      Power
     ⁻          Subtract
 Σ              Sum
Ｉ               Cast to string
                Implicitly print