Consider the following sequence defined as a sum

$$a_n=sum_{k=0}^{n-1}frac{3^{3n-3k-1},(7k+8),(3k+1)!}{2^{2n-2k},k!,(2k+3)!}.$$

QUESTION.For $ngeq1$, is the sequence of rational numbers $a_n$ always integral?

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# Integrality of a sum

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Consider the following sequence defined as a sum

$$a_n=sum_{k=0}^{n-1}frac{3^{3n-3k-1},(7k+8),(3k+1)!}{2^{2n-2k},k!,(2k+3)!}.$$

QUESTION.For $ngeq1$, is the sequence of rational numbers $a_n$ always integral?

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