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Wednesday, March 2, 2022

Hyper-triangular numbers

Triangular number T_n is defined as T_n = 1 + 2 + \dots + n.

Tetrahedral number R_n is defined as R_n = T_1 + T_2 + \dots + T_n.

Let T^k_n denote the k-th dimensional triangular numbers, with T^1_n = 1+1+\dots+1 = n, T^2_n = T_n, and T^3_n = R_n. Specifically, the higher dimensional triangular numbers are defined as: T^{k+1}_n = T^k_1 + T^k_2 + \dots + T^k_n Prove that: T^k_n.T^m_1 + T^k_{n-1}.T^m_2 + \dots + T^k_1.T^m_n = T^{k+m+1}_{n+1}

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