alternating sum with Barnes G functions

Let $G(n)=(n-2)!(n-3)!cdots 1!$ denote the Barnes G-function.
I am pretty sure that
$$
sum_{m=0}^{k^2-1}
(-1)^mbinom{k^2-1}m
frac{G(k+n-m+1)}{G(n-m+1)G(k+1)(k^2)!}
= n-2k^2-2k
$$ when $k$ is odd
and is
$n-frac 12(k^2-1)$ when $k$ is even,
but I lack a proof. I’d already be happy for some references for identities regarding the Barnes function, as this is all new to me.