Pycharmで

from sympy import *
var('i n')
mySum=(summation(i, (i, 1, n)))**2-summation(i**2, (i, 1, n))
print("#        ",         mySum )
print("#expand  ",expand  (mySum))
print("#simplify",simplify(mySum))
print("#",mySum.subs({n:10}))
print("#",mySum.subs({n:100}))
#         -n**3/3 - n**2/2 - n/6 + (n**2/2 + n/2)**2
#expand   n**4/4 + n**3/6 - n**2/4 - n/6
#simplify n*(3*n**3 + 2*n**2 - 3*n - 2)/12
# 2640
# 25164150

from sympy import *
var('i n')
mySum=(Sum(i, (i, 1, n)).doit())**2-Sum(i**2, (i, 1, n)).doit()
print("#        ",         mySum )
print("#expand  ",expand  (mySum))
print("#simplify",simplify(mySum))
print("#",mySum.subs({n:10}))
print("#",mySum.subs({n:100}))
#         -n**3/3 - n**2/2 - n/6 + (n**2/2 + n/2)**2
#expand   n**4/4 + n**3/6 - n**2/4 - n/6
#simplify n*(3*n**3 + 2*n**2 - 3*n - 2)/12
# 2640
# 25164150

参考

summation

sum