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If $s_n=\sum_{k=0}^{n}(-4)^k\binom{n+k}{2k}$ how to prove $s_{n+1}+2s_n+s_{n-1}=0$. One of my student had this question in his exam. Honestly to speak I couldn't get any single idea how to even start. I knew some strategies to find binomial sums but they all couldn't help. It would be great if someone helps.

$$ \mbox{If}\quad s_{n} = \sum_{k = 0}^{n}\left(-4\right)^{k} \binom{n + k}{2k},\quad\mbox{how to prove}\quad s_{n + 1} + 2s_{n} + s_{n - 1} = 0\ ?. $$

• One of my student had this question in his exam.
• Honestly to speak I couldn't get any single idea how to even start.
• I knew some strategies to find binomial sums but they all couldn't help.

It would be great if someone helps.

If $s_n=\sum_{k=0}^{n}(-4)^k\binom{n+k}{2k}$ how to prove $s_{n+1}-2s_n+s_{n-1}=0$. One of my student had this question in his exam. Honestly to speak I couldn't get any single idea how to even start. I knew some strategies to find binomial sums but they all couldn't help. It would be great if someone helps.

If $s_n=\sum_{k=0}^{n}(-4)^k\binom{n+k}{2k}$ how to prove $s_{n+1}+2s_n+s_{n-1}=0$. One of my student had this question in his exam. Honestly to speak I couldn't get any single idea how to even start. I knew some strategies to find binomial sums but they all couldn't help. It would be great if someone helps.