I need the result that for all $t$,


$$\sum\_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i+1} \binom{t-i}{i} C\_{t-i-1} = 0,$$


where $C\_{t-i-1}$ is the $(t-i-1)$-th Catalan number. I've checked for $t$ up to 1000 using Python and the result holds, but I don't really have an intuition for why it would be true. The terms of this sequence are on OEIS ([A068763](https://oeis.org/A068763)) but they're simply called a "generalized Catalan sequence".


Does anyone have a name for this sequence or a citation for this result?



Why is this alternating sum involving Catalan numbers $\sum_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i+1} \binom{t-i}{i} C_{t-i-1} = 0$ for all $t$?
The formula (which holds for $t>1$ but not for $t=1$), is equivalent to
$$\sum\_{t=1}^\infty C\_{t-1}\bigl(x(1-x)\bigr)^t = x,$$
which follows directly from the generating function
$$\sum\_{t=1}^\infty C\_{t-1}x^t = \frac{1-\sqrt{1-4x}}{2}.$$


