A truly puzzling generating function.

Question

Would anyone have any idea on how this is done? Truly puzzled at this point :P

Answer 1

• $a_0 = 2$
• $a_1 = 2\sqrt2 = 2 \dfrac{\sqrt2}{1}$
• $a_2 = \sqrt2(\sqrt2-1) = 2 \dfrac{\sqrt2(\sqrt2-1)}{1 \cdot 2}$
• $a_3 = \dfrac13\sqrt2(\sqrt2-1)(\sqrt2-2) = 2 \dfrac{\sqrt2(\sqrt2-1)(\sqrt2-2)}{1 \cdot 2 \cdot 3}$

$$\begin{array}{rcl} a_n = 2\dbinom{\sqrt 2}{n} \\ f(x) = \displaystyle \sum_{n=0}^{\infty} 2\dbinom{\sqrt 2}{n} x^n \\ f(x) = \displaystyle 2 \sum_{n=0}^{\infty} \dbinom{\sqrt 2}{n} x^n \\ f(x) = \displaystyle 2 (1+x)^{\sqrt2} \\ \end{array}$$