three binomial(n,k) definitions

[Michael Somos]

seqfan,
      I was afraid it would come to this. I just came
to realize that the three natural variations of the
binomial table are actually in use without any real
acknowledgment. Here is the three binomial tables :

      binomial(n,k) according to Maple

n\k  -4   -3   -2   -1    0    1    2    3
   +--------------------------------------
-4 |  1    0    0    0    1   -4   10  -20
-3 | -3    1    0    0    1   -3    6  -10
-2 |  3   -2    1    0    1   -2    3   -4
-1 | -1    1   -1    1    1   -1    1   -1
 0 |  0    0    0    0    1    0    0    0
 1 |  0    0    0    0    1    1    0    0
 2 |  0    0    0    0    1    2    1    0
 3 |  0    0    0    0    1    3    3    1
 
binomial(n,k) according to PARI and Mathematica
 
n\k  -4   -3   -2   -1    0    1    2    3
   +--------------------------------------
-4 |  0    0    0    0    1   -4   10  -20
-3 |  0    0    0    0    1   -3    6  -10
-2 |  0    0    0    0    1   -2    3   -4
-1 |  0    0    0    0    1   -1    1   -1
 0 |  0    0    0    0    1    0    0    0
 1 |  0    0    0    0    1    1    0    0
 2 |  0    0    0    0    1    2    1    0
 3 |  0    0    0    0    1    3    3    1
 
binomial(n,k) = if(k<0|k>n,0,n!/(k!*(n-k)!))

n\k  -4   -3   -2   -1    0    1    2    3
   +--------------------------------------
-4 |  0    0    0    0    0    0    0    0
-3 |  0    0    0    0    0    0    0    0
-2 |  0    0    0    0    0    0    0    0
-1 |  0    0    0    0    0    0    0    0
 0 |  0    0    0    0    1    0    0    0
 1 |  0    0    0    0    1    1    0    0
 2 |  0    0    0    0    1    2    1    0
 3 |  0    0    0    0    1    3    3    1

Notice that for n and k nonnegative, they all agree as
expected. They first two agree when n<0 and k>=0. Now I
think it is obvious that these are all natural and valid
sequence tables. Also obvious that they do not agree for
all integer n and k. Now which should deserve the name
of "binomial coefficients"? More to the point, when the
binomial(n,k) is used in a formula which involves n<0 or
k<0, how are we to guess which of the three interpretations
is to apply? This also relates to my earlier attempt to
raise the issue of two-way infinite and one-way infinite
sequences. Shalom, Michael