Let $$D_{n}=\sum_{j=0}^{n-1}(-1)^{n+j-1}\dfrac{\binom{2n-4}{j}}{n+j-1},E_{n}=\sum_{j=0}^{n-1}(-1)^{n+j-1}\dfrac{\binom{2n-4}{j}}{n+j}$$
I want find $D_{n}$ and $E_{n}$ recursive relations,
I try take somedays,At present I only find following $$\sum_{j=0}^{2n-4}(-1)^j\dfrac{\binom{2n-4}{j}}{n+j-1}=\dfrac{3n-4}{n-1}\sum_{j=0}^{2n-4}(-1)^j\dfrac{\binom{2n-4}{j}}{n+j}$$ can you someone help find the recursive relation with $D_{n}$ and $E_{n}$?
I also post MO
The series $D_{n}$ and $E_{n}$ are of the form \begin{align} D_{n} &= \sum_{j=0}^{n-1} (-1)^{n+j-1} \frac{\binom{2n-4}{j}}{n+j-1} \\ &= \frac{ \binom{2n-4}{n} }{ (2n-1) \Gamma(3n-4) } \, {}_{3}F_{2}(1, 4-n, 2n-1; n+1, 2n; 1) - (-1)^{n} \, \frac{\Gamma(n) \Gamma(2n-3)}{(n-1) \Gamma(3n-4)} \end{align} and \begin{align} E_{n} &= \sum_{j=0}^{n-1} (-1)^{n+j-1} \frac{\binom{2n-4}{j}}{n+j} \\ &= \frac{1}{2n} \, \binom{2n-4}{n} \, {}_{3}F_{2}( 1, 4-n, 2n; n+1, 2n+1; 1) - (-1)^{n} \, \frac{\Gamma(n) \Gamma(2n-3)}{\Gamma(3n-3)}. \end{align} By using the recurrence relations from Wolfram generalized hypergeometric Functions page an attempt can be made to make a recurrence relation for both $D_{n}$ and $E_{n}$.