IMO 1983 LL GBR30

Prove the existence of a unique sequence {un} (n = 0, 1, 2 . . .)

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IMO 1983 LL GBR30

Origin: GBR

Problem

Prove the existence of a unique sequence {un} (n = 0, 1, 2 . . .) of positive integers such that u2 n = n  r=0 n + r r  un−r for all n \geq0, where m r  is the usual binomial coefficient.