Race of the wealthy gamblers: How do I get this closed form? UPDATE:
If you can prove the following identity:
$$\sum\limits_{t=0}^p \left(\frac{{2t+1 \choose t}}{2^{2t+1}}\right)^2 \frac{1}{t+2} = -1 + \frac{(2+p)}{2^{4p+4}}{2p+3 \choose p+1}^2$$
Then this is good enough to solve this question and get my gratitude as well as the 50 point bounty. I got this identity from Mathematica. See my answer below for details.

My question relates to an infinite summation and it's very elegant closed form. The expression is the solution to a nice problem which I'll get into as well. Here is the summation:
Let:
$$a_t = \left({2t+1 \choose t} - {2t+1 \choose t-1}\right)\frac{1}{2^{2+2t}}$$
and,
$$b_t = \left({2t+2 \choose t} - {2t+2 \choose t-1}\right)\frac{1}{2^{3+2t}}$$
For the very first terms of these sequences, ${n \choose -1} = 0$ for all $n$.
And the summation:
$$S = \sum_{t=0}^{\infty} \left(1-\sum_{l=0}^{t-1}b_l\right) a_t = 1- \sum_{t=0}^{\infty} \left(\sum_{l=0}^{t-1}b_l\right) a_t =  7-\frac{20}{\pi} \tag{1}$$
I know the expression above is correct (verified with a python program), but have no idea how to prove this and would like to at least see how I might approach it.
Now, why do I care about this summation? It is the solution to the following problem:

Imagine two wealthy gamblers start tossing their own separate fair coins, winning 1\$ on heads and losing 1\$ on tails. Both start at 0\$ and have infinite bank balances. The first one wants to get to 2\$ and the second one wants to get to 3\$. What is the probability that the first one will reach his goal before the second one? 
One way to solve this is to consider the probability that a gambler reaches exactly $k$ dollars for the first time on toss $n$. If he has $t$ tails, then he needs $t+k$ heads. So, $n=2t+k$ (note if k=2\$, he can only reach the target on even tosses and if k=3\$, he can only reach it on odd tosses). This probability turns out to be:
$$\left({k+2t-1 \choose t} - {k+2t-1 \choose t-1}\right) \frac{1}{2^{k+t}} \frac{1}{2^t}$$
Now, let $A_n$ be the probability that the 2\$ targeting gambler wins on toss $n$ and $A$ be the probability that he wins. Then we have $A = \bigcup\limits_{n=1}^{\infty}A_n$ and so, $P(A) = \sum\limits_{n=0}^\infty P(A_n)$. Now, for the 2\$ targeting gambler to win on the $n$th toss, two things should happen:


*

*He should reach his target on the $n$th toss for some even $n$.

*His competitor, the 3\$ gambler should not reach his target on any toss upto $n-1$ (since he can only reach his target on odd tosses).


Putting all of this together, you can see that the probability that the 2\$ gambler wins is given by equation (1) above. I have put together some python code that approximates $S$ by going upto a large number of tosses. A Reddit user pointed out the closed form for which he used a slightly different but related approach and Mathematica. Now, how do I prove that the summation above has the closed form mentioned $(7-\frac{20}{\pi})$?
EDIT: 
Here is a short python snippet that demonstrates the summation in equation (1) above.
a_t = np.array([(comb(2*t+1,t)-comb(2*t+1,t-1))/2**(2*t+2) for t in range(500)])
b_t = np.array([(comb(2*t+2,t)-comb(2*t+2,t-1))/2**(2*t+3) for t in range(500)])
b_sum = 1-np.concatenate(([0],np.cumsum(b_t)))
s = sum(a_t*b_sum[:500])
print(7-20/np.pi-s)

Also, here is the Mathematica snippet that shows the result (thanks to @SteveKass for helping with that):

 A: As Steve Kass commented, there is a problem somewhere.
Using a CAS, what I obtained after simplifications is 
$$\sum_{l=1}^{t-1}b_l=\frac{7}{8}+\frac{ (t+3) (3 t+4) }{3
   (t+1)\,2^{2 (t+1)}}\left(\binom{2 t+2}{t-1}-\binom{2 t+2}{t}\right)=\frac{7}{8}-\frac{(3 t+4)\, \Gamma \left(t+\frac{3}{2}\right)}{\sqrt{\pi } \,\,\Gamma (t+3)}$$ The partial sum
$$S_p = \sum_{t=1}^{p} \left(\sum_{l=1}^{t-1}b_l\right) a_t$$ is evaluated (the result is very nasty but "almost" explicit ) but,  given by the same CAS,
$$S_\infty =\frac{20}{\pi }-\frac{195}{32}$$ is obtained without any problem (how ? this is the question).
Edit
After your two major changes (summations starting at $0$ and the formula for $S$), the results become quite different.
$$\sum_{l=0}^{t-1}b_l=1-\frac{(3 t+4) \Gamma \left(t+\frac{3}{2}\right)}{\sqrt{\pi } \,\,\Gamma (t+3)}$$
$$S_p = \sum_{t=0}^{p} \left(1-\sum_{l=0}^{t-1}b_l\right) a_t=7-\frac{4 (5 p+12) \,\Gamma \left(p+\frac{5}{2}\right)^2}{\pi \, \Gamma (p+3)^2}$$
Using Stirling approximation for lage values of $p$, we have
$$S_p=7-\frac{20}\pi\left(1+\frac{3}{20 p}-\frac{59}{160 p^2}+\frac{573}{640 p^3}-\frac{21977}{10240
   p^4}+O\left(\frac{1}{p^5}\right)\right)$$
$$S_\infty =7-\frac{20}{\pi }$$
A: The basic idea of this question was to understand how $\pi$ was coming into these expressions for competing gambler problems. So, I decided to tackle a slightly simpler problem. Instead of finding the probability a 2\$ chasing gambler would beat a 3\$ chasing gambler, I tried to find the probability that a 2\$ chasing gambler would beat a 1\$ chasing gambler. The answer here is $4/\pi-1$. 
First, I wrote the PDF of the stopping time of the 1\$ chasing gambler in a simplified form:
$$z_t = \frac{2t \choose t}{(t+1)2^{2t+1}}$$
Then using Mathematica, I got the survival function (there is also an elegant way to get it from "scratch"):
$$Z_t = \sum\limits_{l=t+1}^{\infty}z_t = \frac{2t+1 \choose t}{2^{2t+1}}$$
Similarly, the PDF of the stopping time of the 2$ chasing gambler becomes:
$$a_t = \frac{2t+1 \choose t}{(t+2)2^{2t+1}}$$
The probability the 2\$ targeting gambler will beat the 1\$ targeting gambler in $2p+2$ tosses then becomes:
$$S_p = \sum\limits_{t=0}^{p} a_t Z_t = \sum\limits_{t=0}^{p} \frac{{2t+1 \choose t}^2}{(t+2)2^{4t+2}}$$
Now, this is the part that I haven't figured out how to do without Mathematica. The expression for $S_p$ becomes:
$$S_p = -1 + (p+2)\frac{{2p+3 \choose p+1}^2}{2^{4p+4}} \tag{2}$$
Now, using the sterling approximation for the factorials and letting $p \to \infty$ we get:
$$S_{\infty} = \frac{4}{\pi}-1$$
Everything is fine apart from equation (2). Don't know how to prove that one. Which is why I added an update to the question on the very top asking for help with proving this expression.
But, I'm starting to get a rough inkling for where these $\pi$'s are coming from. Basically, all these expression describing the PDF's and survival functions of the gamblers involve binomial terms close to the mode. 
The probabilities we're after (one of the gamblers beating the other) involve two of these binomial terms close to their mode multiplied over all possible $t$ up to $p$. These summations tend to produce expressions that also involve squares of binomial terms close to the mode, this time involving $p$. And this is the part I need help with. I don't understand why this happens. 
However once we accept that, we'll be left with some term like equation (2) which will involve a binomial coefficient around the mode for a large $p$. However as $p$ becomes large, this binomial distribution starts to resemble a Gaussian (law of large numbers). And we know that the gaussian around it's mode has a PDF with a $\frac{1}{\sqrt{\pi}}$ term. So, squaring it yields a $\frac{1}{\pi}$ term. 
A: We have the identity,
$$
\begin{align}
\frac{k+1}{16^{k+1}}\binom{2k+2}{k+1}^2-\frac{k}{16^k}\binom{2k}{k}^2
&=\frac1{16}\frac{k+1}{16^k}\binom{2k}{k}^2\left(\frac{4k+2}{k+1}\right)^2-\frac{k}{16^k}\binom{2k}{k}^2\\
&=\frac1{16^k}\binom{2k}{k}^2\left(\frac{k+1}{16}\left(\frac{4k+2}{k+1}\right)^2-k\right)\\
&=\frac1{16^k}\binom{2k}{k}^2\frac1{4(k+1)}\tag1
\end{align}
$$
Applying $(1)$:
$$
\begin{align}
\sum_{k=0}^p\left(\frac{\binom{2k+1}{k}}{2^{2k+1}}\right)^2\frac1{k+2}
&=\sum_{k=1}^{p+1}\left(\frac{\binom{2k-1}{k-1}}{2^{2k-1}}\right)^2\frac1{k+1}\\
&=\sum_{k=1}^{p+1}\left(\frac{\binom{2k}{k}}{2^{2k}}\right)^2\frac1{k+1}\\
&=4\sum_{k=1}^{p+1}\left(\frac{k+1}{16^{k+1}}\binom{2k+2}{k+1}^2-\frac{k}{16^k}\binom{2k}{k}^2\right)\\
&=4\frac{p+2}{16^{p+2}}\binom{2p+4}{p+2}^2-1\\
&=\frac{p+2}{16^{p+1}}\binom{2p+3}{p+1}^2-1\tag2
\end{align}
$$

