restart; interface(version);
  Classic Worksheet Interface, Maple 12.02, Windows, Dec 10 2008 Build ID 377066
B:=binomial(x,k)/binomial(-x,k);
convert(B, GAMMA);
simplify(%);
% assuming x::integer;
                                  0
#but
subs(x=1, B); subs(k=1, %); 
simplify(%);
                                  -1
# may be related?
S:=sin(Pi*k)/Pi/k;
S assuming k::integer;
                                  0

limit(S, k=0);
                                  1

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