I think this illustrates the problem more clearly; possibly related to a complaint that I got bounced on :)
def H(n,k): return hypergeometric([-k, -n + k], [-k], -1)
def Ha(n,k): return hypergeometric([ -n + k], [], -1)
def T(n,k): return int(H(n,k).n())
def Ta(n,k): return int(Ha(n,k).n())
def S(n,k): return H(n,k).simplify_hypergeometric()
def Sa(n,k): return Ha(n,k).simplify_hypergeometric()
for n in range(8):
print('(n,k),T(n,k) ' + ', '+str([((n,k),T(n, k)) for k in
range(n+1)]))
print('(n,k),Ta(n,k)' + ', '+str([((n,k),Ta(n, k)) for k
in range(n+1)]))
print('(n,k),S(n,k) ' + ', '+str([((n,k),S(n, k)) for k in
range(n+1)]))
print('(n,k),Sa(n,k)' + ', '+str([((n,k),Sa(n, k)) for k
in range(n+1)]))
print('--------------')
The result is attached; since it goes on for a while. The core is that the numerator [-k] and denominator [-k] is not always cancelled. Apparently the problem reaches further than I thought.
Ray
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