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Counting & ProbabilityHardIdentities and Techniques, Proved

Identities and Techniques, Proved · Hard problem

Given 13+23+⋯+n3=(n+12)21^3+2^3+\cdots+n^3=\binom{n+1}{2}^2, find the sum of the cubes of the odd numbers, 13+33+53+⋯+1931^3+3^3+5^3+\cdots+19^3.