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IMO ShortList 1998, algebra problem 4

Source: IMO ShortList 1998, algebra problem 4; Polish 1st round, 1999

October 22, 2004
functioncombinatoricscountingsymmetrybinomial coefficientsIMO Shortlist

Problem Statement

For any two nonnegative integers nn and kk satisfying n≥kn\geq k, we define the number c(n,k)c(n,k) as follows: - c(n,0)=c(n,n)=1c\left(n,0\right)=c\left(n,n\right)=1 for all n≥0n\geq 0; - c(n+1,k)=2kc(n,k)+c(n,k−1)c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right) for n≥k≥1n\geq k\geq 1. Prove that c(n,k)=c(n,n−k)c\left(n,k\right)=c\left(n,n-k\right) for all n≥k≥0n\geq k\geq 0.
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