<s>[INST] Is it possible to write Catalan's product for e as a product of a sequence? Catalan found a product for $e$:
$$e=\dfrac{2}{1}\left(\dfrac{4}{3}\right)^{\frac{1}{2}}\left(\dfrac{6\cdot 8}{5\cdot 7}\right)^{\frac{1}{4}}\left(\dfrac{10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}\right)^{\frac{1}{8}}\cdots$$
Is it possible to write this as an infinite product in the form:
$$e = \prod_{n=0}^\infty a_n$$
I know that it would likely be some sequence of factorials to the $\frac{1}{2^n}$ power, but I cannot think of what the sequence would be.
Thanks!
 [/INST] Look at that fourth term:
$$\left(\frac{10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}\right)^{1/8}$$
Factor out the $2$s in the numerator:
$$10\cdot 12\cdot 14\cdot 16=2^4\cdot5\cdot6\cdot7\cdot 8=2^4\frac{8!}{4!}$$
The denominator is $$\begin{align}9\cdot 11\cdot 13\cdot 15 &= \frac{16!}{2^8\cdot 8!\cdot (1\cdot 3\cdot 5\cdot 7)}\\
1\cdot 3\cdot 5\cdot 7&=\frac{8!}{2^4 4!}
\end{align}$$
So:
$$9\cdot 11\cdot 13\cdot 15 = \frac{16!\cdot 2^4\cdot 4!}{2^8\cdot 8!\cdot 8!}=\binom{16}{8}\frac{4!}{2^4}$$
So $$a_4^{8}=\frac{2^4\frac{8!}{4!}}{\binom{16}{8}\frac{4!}{2^4}}=\frac{2^8\binom{8}{4}}{\binom{16}{8}}$$
of $$a_4=2\left(\frac{\binom{8}{4}}{\binom{16}{8}}\right)^{1/8}$$
Thus, for $n>1$, we might guess:
$$a_n =2\left(\frac{\binom{2^{n-1}}{2^{n-2}}}{\binom{2^n}{2^{n-1}}}\right)^{1/2^{n-1}}$$
Somewhat tricky, because $n=1$ doesn't work.
If you want to expand it as factorials:
$$a_n==2\left(\frac{\left(2^{n-1}!\right)^3}{2^n!\left(2^{n-2}!\right)^2}\right)^{1/2^{n-1}}$$
If you define $$b_k=2\left(\frac{\left((2k)!\right)^3}{(4k)!(k!)^2}\right)^{\frac{1}{2k}}$$
Then $a_n=b_{2^{n-2}}$ for $n\geq2$ and $a_1=1$.
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