Split the entire series into two parts

(1*2*3+3*4*5+.........)-(2^3+4^3+.......)

The first part can be written as Summation[n(n+1)(n+2)]

Summation (n^3 + 3(n^2) + 2n)

I hope you can expand for the individual summations.

For the second part take 2^3 common

2^3(1+2^3 + 3^3......)

2^3 Summation(n^3)

Hope you can take it from here.