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In Sage:

binomial(-1,-1) = 0

I have complaint about this before: ask-sage and proposed the natural binomial(x,x) = 1 for all x.

I discussed the arguments in detail at sagemath-track where I opened a ticket.

One answer was: "Having binomial(z, z) != 1 is collateral damage."

There is also the damage of inconsistency.

For me inconsistency is what this code shows:

PartitionCoefficient = lambda p: mul(binomial(p[j], p[j+1]) for j in range(len(p)-1))
for n in (0..6):
    for k in (0..n):
        P = Partitions(n, max_part=k, inner=[k])
        print sum(PartitionCoefficient(p) for p in P), binomial(n-1,k-1)  

1 0
0 0
1 1
0 0
1 1
1 1
0 0
1 1
2 2
1 1
0 0
1 1
3 3
3 3
1 1
0 0
1 1
4 4
6 6
4 4
1 1
0 0
1 1
5 5
10 10
10 10
5 5
1 1

Another said: "Apparently Maple and Mathematica use the proposed version. Is that sufficient justification to change our version?"

Question: Is the internal consistency of Sage enough justification?