Hurray! Maple's 2023 update arrived!

[peter....@gmail.com]

T := (n, k) -> -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2):
seq(print(seq(simplify(T(n, k)), k=1..n)), n=1..6);

Maple 2020.2 says:

1
2, 2
3, 6, 3
4, 12, 13, 4
5, 20, 34, 24, 5
6, 30, 70, 80, 40, 6

Maple 2023.2 says:
Error, (in GAMMA) numeric exception: division by zero

Who is right?

[acer]

I have submitted a bug report.

I suspect that the "exception" occurs due to un-careful handling of GAMMA (ostensibly in denominator). I note that even in older versions there is the following inconsistency:

simplify(2*hypergeom([1, 5], [4], 2));
-1
simplify(hypergeom([1, 5], [4], 2));

Error, (in GAMMA) numeric exception: division by zero

Not all the hypergeom calls in your example present the problem. There are a few workarounds. For example,

restart;

T := (n, k) -> -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2):

seq(print(seq(eval(simplify(T(nn,k)),nn=n), k=1..n)), n=1..6);

1
2, 2
3, 6, 3
4, 12, 13, 4
5, 20, 34, 24, 5
6, 30, 70, 80, 40, 6

Or, (expensive way to handle this example involving non-convergent sum and analytic continuation of the hypergeometric function...),

sum(op(combine(convert(hypergeom([1,5],[4],2),compose,MeijerG,Sum))),'formal')
-1/2