Via http://www.wolframalpha.com , an admittedly unreliable source ... In: (Binomial[n-1, k-1] + Binomial[n-1, k] - Binomial[n, k]) Out: 0 In: (Binomial[n-1, k-1] + Binomial[n-1, k] - Binomial[n, k]) /. {n -> 0, k -> 0} Out: 1 Oops! Maple, on the other hand, discreetly if uninformatively desists from further simplification of the general expression. But not merely from cowardice: given the Catalan number recurrence, which Wolfa either claims implausibly not to understand, or else goes all coy over and refuses to reply at all: In: (Binomial[n+n, n]/(n+1) - Sum[(Binomial[j+j-2, j-1]/j)(Binomial[n+n-j-j, n-j]/(n-j+1)), {j, 1, n}]) Out: <caposcalpitation / mysterious silence> Maple in contrast fearlessly extends its neck: In: n := 'n'; simplify( binomial(n+n, n)/(n+1) - sum(binomial(j+j-2, j-1)/j*binomial(n+n-j-j, n-j)/(n-j+1), j = 1..n) ); Out: 0 In: n := 0; binomial(n+n, n)/(n+1) - add(binomial(j+j-2, j-1)/j*binomial(n+n-j-j, n-j)/(n-j+1), j = 1..n); Out: 1 and duly measures its length on the carpet (which one way and another seems to be having a hard time of it lately). Would you buy a used hypergeometric identity from any of these head-CASes? Fred Lunnon