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Let $B(n)=(n-2)!(n-3)!\cdots 1!$ denote the Barnes G-function. I am pretty sure that $$ \sum_{m=0}^{k^2-1} (-1)^m\binom{k^2-1}m \frac{G(k+n-m+1)}{G(n-m+1)G(k+1)(k^2)!} = n-2k^2-2k $$ 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.

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What is $G(n)$? – Vladimir Reshetnikov May 10 '13 at 6:34
Sorry, B(n)=G(n) - typo. – JM Landsberg May 10 '13 at 22:47

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