All those couples of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which do not depends on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)_{-n}$ as $\frac{(-1)^n}{(1-a)^n}$ if $a \neq 1$, and $(1)_{-n}$ as $\frac{n(-1)^n}{(1)_n}$ which preserves formally the recurrence $\Gamma(x+1)=x \, \Gamma(x)$ (see Chapter 7 of the book A=B by Petkovsek, Wilf, Zeilberger) and another application to the WZ-method in the Section 4 of [this paper][1]). The very nice formula for $\zeta(5)$ discovered by zy_ allow us to discover a new "divergent" (convergent by analytic continuation) Ramanujan-like series for $1/\pi^4$ by using the upside-down-transformation. 
 
The transformation $n \to n+1/2$ applied to $\lambda(n)$ essentially inverts $\lambda(n)$ giving $\frac{1}{\lambda(n)}$ but do not invert $z^n$. Hence it is not an upside-down transformation. This explains why the pattern observed in the post cannot be generalized in the way pointed out by the author.   

In the Appendix of [this paper][2] there are examples of the "upside-down" technique and in [this unpublished file][3] there are many examples of the transformation $n  \to n + \frac12$.


  [1]: https://arxiv.org/pdf/1012.2681.pdf
  [2]: https://arxiv.org/pdf/1610.04839.pdf
  [3]: https://drive.google.com/file/d/0B_wpzKF9_poyMU5jWFZfbHlsZjA/view