Sum formulas of multiple zeta values with even arguments and polynomial weights

The special caseE0(2,n,k)=∑α1+⋯+αk=nζ(2α1,2α2,⋯,2αk) for positive integers α1,α2,⋯,αr with αr≥2 was evaluated by Hoffman [7] in 2017 as(−1)n−k−1π2n(2n+1)!∑i=0n−k(n−ik)(2n+12i)(22i−2)B2i, where Bn (n=0,1,2,⋯) are Bernoulli numbers.