Python 3, 28 bytes

lambda n:n*~-n*-~n*(n/4+1/6)

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Implements \$n(n-1)(n+1)(3n+2)/12\$

Python 2, 29 bytes: lambda n:n*~-n*-~n*(3*n+2)/12

Python 3,  28  27 bytes

-1 thanks to xnor

lambda n:(n**3-n)*(n/4+1/6)

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Implements \$n(n-1)(n+1)(3n+2)/12\$

Python 2,  29  28 bytes: lambda n:(n**3-n)*(3*n+2)/12

Python 3, 28 bytes

(Assuming code-golf is the winning criteria, but also efficient at O(1).)

lambda n:n*~-n*-~n*(n/4+1/6)

Try it online!

Python 2, 29 bytes: lambda n:n*~-n*-~n*(3*n+2)/12

Python 3, 28 bytes

lambda n:n*~-n*-~n*(n/4+1/6)

Try it online!

Implements \$n(n-1)(n+1)(3n+2)/12\$

Python 2, 29 bytes: lambda n:n*~-n*-~n*(3*n+2)/12

Python 2, 29 bytes

(Assuming code-golf is the winning criteria, but also efficient at O(1).)

lambda n:n*~-n*-~n*(3*n+2)/12

Try it online!

Python 3, 28 bytes

(Assuming code-golf is the winning criteria, but also efficient at O(1).)

lambda n:n*~-n*-~n*(n/4+1/6)

Try it online!

Python 2, 29 bytes: lambda n:n*~-n*-~n*(3*n+2)/12