Suppose $I_{2n}=\int_0^{\pi}x\sin^{2n}x \;dx$. Then one can obtain the following: $$I_0= \frac{\pi ^2}{2}, I_1= \frac{\pi ^2}{4}, I_2= \frac{3\pi ^2}{16}, I_3= \frac{5\pi ^2}{32},\ldots, I_n= \frac{{2n\choose n}\pi ^2}{2^{2n+1}}. $$
Note that the Maclurin series of $\frac{1}{\sqrt{1-x}}$ is
$$1+\frac{x}{2}+\frac{3x^2}{8}+\frac{5x^3}{16}+\ldots + \frac{(-1)^n{2n \choose n}x^n}{2^{2n}}+\ldots.$$
This means thatthe Maclurin series of $\frac{1}{\sqrt{1-x}}$ is $$\frac{2}{\pi ^2}\sum_{n=0}^\infty I_nx^n$$
Is there any relationship, in general, for other recurrence relation and coefficients of Maclaurin series? If yes, what is that?
For a general discussion of recurrences and series take a look at Wilf's "generatingfunctionology".