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I think that Mathematica says that binomial(-1, 0) is 1, as is binomial(-1, -1). (They may be taking their extension to negative entries from http://arxiv.org/pdf/1105.3689.pdf?) So if I'm right, then the identity
binomial(n,k) + binomial(n, k+1) = binomial(n+1, k+1)
fails when n=-1, k=-1. Sage seems to use a different convention so that this identity is preserved, or at least that's my guess as to what's going on:
sage: binomial(-1, -1)
0
sage: binomial(-1, 0)
1
sage: binomial(-1, -1) + binomial(-1, 0) == binomial(0, 0)
True
See http://en.wikipedia.org/wiki/Pascal%27s_triangle#Extensions.
