Some weighted sum identities for double zeta values

We obtain the weighted sum identities for

$$\begin{gathered} \sum\limits_{k = 2}^{s - 1} {{{\left( { - 1} \right)}^k}k\varsigma \left( {k,s - k} \right)} ,\sum\limits_{k = 1}^{s - 1} {k\varsigma \left( {2k,2s - 2k} \right),} \sum\limits_{k = 1}^{s - 1} {k\varsigma \left( {2k + 1,2s - 2k - 1} \right),} \hfill \\ \sum\limits_{k = 1}^{s - 1} {{k^2}\varsigma \left( {2k,2s - 2k} \right){\kern 1pt} and{\kern 1pt} \sum\limits_{k = 1}^{s - 1} {{k^2}\varsigma \left( {2k + 1,2s - 2k - 1} \right).} } \hfill \\ \end{gathered} $$

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