<s>[INST] What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$ For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series
$$
\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n.
$$
For $b=0$, this post shows
$$
\sum_{n=0}^\infty \binom{2n}{n}x^n = (1-4x)^{-1/2}.
$$
How do we proceed for an integer $b>0$?
I tried to start from the power series of
$$
(1-x)^{-(b+2n)},
$$
but it doesn't work.
 [/INST] Note  that the  quoted identity  is not  difficult to  verify  using a
variant of Lagrange Inversion. Introduce
$$T(z) = w = \sqrt{1-4z}$$ so that
$$z = \frac{1}{4} (1-w^2)$$ and
$$dz = -\frac{1}{2} w \; dw$$
Then we seek to compute
$$[z^n] \frac{2^b}{T(z)} (1+T(z))^{-b}
= \frac{1}{2\pi i} 
\int_{|z|=\epsilon} \frac{1}{z^{n+1}}
\frac{2^b}{T(z)} (1+T(z))^{-b} \; dz.$$
Using the substitution this becomes
$$- \frac{1}{2\pi i}
\int_{|w-1| = \epsilon} \frac{4^{n+1}}{(1-w^2)^{n+1}}
\frac{2^b}{w} (1+w)^{-b} \frac{1}{2} w \; dw
\\ = - \frac{1}{2\pi i}
\int_{|w-1| = \epsilon} 
\frac{4^{n+1}}{(1-w)^{n+1} \times (1+w)^{n+1+b}}
2^{b-1} \; dw
\\ = - \frac{1}{2\pi i}
\int_{|w-1| = \epsilon} 
\frac{(-1)^{n+1} 4^{n+1}}{(w-1)^{n+1} \times (1+w)^{n+1+b}}
2^{b-1} \; dw.$$
It follows that the value of the integral is given by
$$(-1)^n 4^{n+1} 2^{b-1}
[(w-1)^n] \frac{1}{(1+w)^{n+1+b}}
\\ = (-1)^n 4^{n+1} 2^{b-1}
[(w-1)^n] \frac{1}{(2+(w-1))^{n+1+b}}
\\ = (-1)^n 4^{n+1} 2^{b-1} \frac{1}{2^{n+1+b}}
[(w-1)^n] \frac{1}{(1+(w-1)/2)^{n+1+b}}
\\ = (-1)^n  2^{2n+b+1} \frac{1}{2^{n+1+b}}
\frac{(-1)^n}{2^n} {n+n+b\choose n+b}
\\= {2n+b\choose n+b}.$$
</s>