This is a bug in SumSum
.
In addition to the nice workarounds proposed by others, the substitution m = 2p$m = 2p$ also leads to a correct answer for this example.
In[1]:= Sum[ Binomial[2 p - 2, k - 1] (k - 1), {k, 2, 2 p - 2, 2}] /. {p -> m/2} // Simplify
Out[1]= 2^(-4 + m) (-2 + m)
In[1]:= Sum[
Binomial[2 p - 2, k - 1] (k - 1), {k, 2, 2 p - 2, 2}] /. {p ->
m/2} // Simplify
Out[1]= 2^(-4 + m) (-2 + m)
Sorry for the inconvenience caused by this problem.