Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for
![$\zet (4n+3)$](/math_tex/cf343eea482166ab88e5b1e69c76170982.gif)
that generalizes Apéry's series for
![$\zet$(3)$](/math_tex/012bb81e53c763080d73b45ce0d53d4182.gif)
, and appears to give the best possible series relations of this type. The formula reduces to a finite but apparently nontrivial combinatorial identity. The identity is equivalent to an interesting new integral evaluation for the central binomial coefficient. We outline a new technique for transforming and summing certain infinite series. We also derive a formula that provides strange evaluations of a large new class of nonterminating hypergeometric series.