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In this paper, we give a proof of a conjecture made by Zagier about theinverse of some matrix related to double zeta values of parity$(\mathrm{even},\mathrm{odd})$. As a result, we obtain a family of Bernoullinumber identities. We further generalize this family to a more general settinginvolving binomial coefficients of negative arguments.

Inverse of a matrix related to double zeta values of odd weight