What is the name of this sequence that approximates $\pi$? I came across this sequence $$\sum_{n=1}^{\infty}\frac{2^n}{2n\choose n}n^m=a_{m}+b_{m}\pi$$ where $a_m$ and $b_m$ are rational with $\lim_{m\to\infty}\frac{a_m}{b_m}=\pi$. I cannot figure out where it comes from or if it has a name. Any ideas?

Write it as $$S_m=\sum_{n=1}^{\infty}\frac{2^n}{2n\choose n}n^m=a_{m}+\frac \pi 2 \, b_{m}$$
The first $a_m$ are
$$\{1,3,11,55,355,2807,26259,283623,3473315,47552791,719718067\}$$ and this is sequence $A180875$ in $OEIS$.
Quoting $OEIS$, these are "Lehmer's coefficients stemming from an inverse binomial coefficient series"
The first $b_m$ are
$$\{1,2,7,35,226,1787,16717,180560,2211181,30273047,458186752\}$$ and this is sequence $A014307$ in $OEIS$.
If you look in the formula sections, you will find approximations given in year $2014$ by Vaclav Kotesovec, namely
$$a_m \sim \frac {\pi  m^{m+1} }{\sqrt{2} e^m \log ^{m+\frac{3}{2}}(2) }\qquad \qquad b_m\sim\frac{\sqrt{2} (m+1)^{m+1}}{e^{m+1} \log ^{m+\frac{3}{2}}(2) }$$
So, using these approximations
$$2\frac { a_m}  {b_m}\sim \pi \,e \,\left(\frac m {m+1} \right)^{m+1}$$ which is not good for an estimation of the error.
For $0 \leq m \leq 15$, a quick and dirty linear regression gives $(R^2=0.979)$ gives
$$\log \left(\left| 2\frac{ a_m}{b_m}-\pi \right| \right)=-2.14718 m-1.37498$$ Increasing the range of $m$ and searching for nice looking numbers in inverse symbolic calculators
$$\left| 2\frac{ a_m}{b_m}-\pi \right| \sim e^{-e/2} \left(\frac{\log (2)}{2 \pi }\right)^m$$
Edit
Formally, in terms of  the generalized hypergeometric function
$$S_m=\, _{(m+1)}F_m\left(2,\cdots,2;1,\cdots,1,\frac{3}{2};\frac{1}{2}\right)$$
