A short proof of a non-vanishing result by Conca, Krattenthaler and Watanabe (Q6082486)

In that paper, the author wrote a short proof of a non-vanishing result by \textit{A. Conca} et al. [Rend. Semin. Mat. Univ. Padova 121, 179--199 (2009; Zbl 1167.05051)]. He proposed a shorter and elementary proof, based on the following Theorem. \textbf{Theorem 1.} For any \(h\geq 1\), consider the polynomials \[ a_{h}:=\sum_{b=0}^{\left\lfloor h/3\right\rfloor }\frac{(-1)^{h-b}}{h-b} \binom{h-b}{2b}U^{b}\in \mathbb{Q}[U] \] and \(sh:=h\cdot a_{h}\). Then the sequence \((s_{h})\) for \(h\geq 1\) satisfies the linear recurrence \[ s_{h+3}+2s_{h+2}+s_{h+1}=U\cdot s_{h} \] for all \(h\geq 1\). Further he deduced the following Corollary. \textbf{Corollary 2.} For any \(h\geq 1\), the rational number \[ \sum_{b=0}^{\left\lfloor h/3\right\rfloor }\frac{(-1)^{h-b}}{h-b}\binom{h-b}{ 2b}(\frac{2}{3})^{b} \] is zero except for \(h=3\).