I am not sure whether an infinite product whose factors are fractions qualifies under the restriction "explicit rational sequence". In case it does, the following should be relevant.

A useful resource for identifying relevant sequences is the OEIS (On-line Encyclopedia of Integer Sequences). In this case, one could search it for sqrt(2), which finds myriad sequences somehow related to or involving $\sqrt{2}$. Since I had an idea what I might be looking for, I relatively quickly identified an infinite product whose factors are fractions, with the numerators specified by sequence A016826 and the denominators specified by sequence A001539. I then used Wolfram Alpha to verify that

$$\prod_{n=0}^{\infty}\frac{(4n+2)^{2}}{(4n+1)(4n+3)} = \prod_{n=1}^{\infty}\frac{1}{1-\frac{1}{4(2n-1)^{2}}} = \frac{4}{3}\cdot\frac{36}{35}\cdot\frac{100}{99}\cdot\frac{196}{195}\cdot \ldots\ = \sqrt{2}$$

An alternate approach to identifying similar infinite products would be to search the literature with Google Scholar (or just plain Google) for Wallis-type infinite products.

I am not sure whether an infinite product whose factors are fractions qualifies under the restriction "explicit rational sequence". In case it does, the following should be relevant.

A useful resource for identifying relevant sequences is the OEIS (On-line Encyclopedia of Integer Sequences). In this case, one could search it for sqrt(2), which finds myriad sequences somehow related to or involving $\sqrt{2}$. Since I had an idea what I might be looking for, I relatively quickly identified an infinite product whose factors are fractions, with the numerators specified by sequence A016826 and the denominators specified by sequence A001539. I then used Wolfram Alpha to verify that

$$\prod_{n=0}^{\infty}\frac{(4n+2)^{2}}{(4n+1)(4n+3)} = \prod_{n=1}^{\infty}\frac{1}{1-\frac{1}{4(2n-1)^{2}}} = \frac{4}{3}\cdot\frac{36}{35}\cdot\frac{100}{99}\cdot\frac{196}{195}\cdot \ldots\ = \sqrt{2}$$

An alternate approach to identifying similar infinite products would be to search the literature with Google Scholar (or just plain Google) for Wallis-type infinite products.

I am not sure whether an infinite product whose factors are fractions qualifies under the restriction "explicit rational sequence". In case it does, the following should be relevant.

A useful resource for identifying relevant sequences is the OEIS (On-line Encyclopedia of Integer Sequences). In this case, one could search it for sqrt(2), which finds myriad sequences somehow related to or involving $\sqrt{2}$. Since I had an idea what I might be looking for, I relatively quickly identified an infinite product whose factors are fractions, with the numerators specified by sequence A016826 and the denominators specified by sequence A001539. I then used Wolfram Alpha to verify that

$$\prod_{n=0}^{\infty}\frac{(4n+2)^{2}}{(4n+1)(4n+3)} = \prod_{n=1}^{\infty}\frac{1}{1-\frac{1}{4(2n-1)^{2}}} = \frac{4}{3}\cdot\frac{36}{35}\cdot\frac{100}{99}\cdot\frac{196}{195}\cdot \ldots\ = \sqrt{2}$$

I am not sure whether an infinite product whose factors are fractions qualifies under the restriction "explicit rational sequence". In case it does, the following should be relevant.

A useful resource for identifying relevant sequences is the OEIS (On-line Encyclopedia of Integer Sequences). In this case, one could search it for sqrt(2), which finds myriad sequences somehow related to or involving $\sqrt{2}$. Since I had an idea what I might be looking for, I relatively quickly identified an infinite product whose factors are fractions, with the numerators specified by sequence A016826 and the denominators specified by sequence A001539. I then used Wolfram Alpha to verify that

$$\prod_{n=0}^{\infty}\frac{(4n+2)^{2}}{(4n+1)(4n+3)} = \prod_{n=1}^{\infty}\frac{1}{1-\frac{1}{4(2n-1)^{2}}} = \frac{4}{3}\cdot\frac{36}{35}\cdot\frac{100}{99}\cdot\frac{196}{195}\cdot \ldots\ = \sqrt{2}$$

An alternate approach to identifying similar infinite products would be to search the literature with Google Scholar (or just plain Google) for Wallis-type infinite products.