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sum over long fraction

Source: Vietnam TST 1994 for the 35th IMO, problem 6

June 25, 2005
calculusderivativefunctioncombinatorics unsolvedcombinatorics

Problem Statement

Calculate T=∑1n1!⋅n2!⋅⋯n1994!⋅(n2+2⋅n3+3⋅n4+…+1993⋅n1994)!T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!} where the sum is taken over all 1994-tuples of the numbers n1,n2,…,n1994∈N∪{0}n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\} satisfying n1+2⋅n2+3⋅n3+…+1994⋅n1994=1994.n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994 \cdot n_{1994} = 1994.
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