alternating sum with Barnes G functions

MathOverflow Asked by JM Landsberg on September 21, 2020

Let $G(n)=(n-2)!(n-3)!cdots 1!$ denote the Barnes G-function.
I am pretty sure that
= 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.

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