MHB What is the result of the sum of binomial coefficients with alternating signs?

AI Thread Summary
The discussion focuses on evaluating the sum of binomial coefficients with alternating signs, specifically the expression S = ∑(-1)^k {2n choose k} {4n choose 2k}. A proposed formula for this sum is S = (-1)^n (6n)!(2n)! / (4n)!(3n)!n!, which was supported by calculations for specific values of n and verified against the OEIS. Participants in the thread confirm the correctness of the proposed formula and provide a hint involving polynomial identities to aid in proving it. The conversation highlights the predictive value of the formula based on calculated results for various n. The thread concludes with encouragement for further exploration of the proof.
hxthanh
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Evaluate sum:

$\displaystyle S=\sum_{k=0}^{2n}(-1)^k{2n\choose k}{4n\choose 2k}$
 
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hxthanh said:
Evaluate sum:

$\displaystyle S_n=\sum_{k=0}^{2n}(-1)^k{2n\choose k}{4n\choose 2k}$
I believe that the answer must be $$S = \frac{(-1)^n(6n)!(2n)!}{(4n)!(3n)!n!}$$, but I have NO idea how one might prove that.

What I did was to calculate $S_n$ when $n=1,\,2,\,3$, and then look those numbers up in the OEIS. This gave the above formula, and suggested that the next result, when $n=4$, would be $104006$. I then laboriously calculated $S_4$, and found that it is indeed $104006$. So my proposed formula has predictive value and therefore must be correct. (Wasntme) Now perhaps somebody can tell me why.
 
Opalg said:
I believe that the answer must be $$S = \frac{(-1)^n(6n)!(2n)!}{(4n)!(3n)!n!}$$, but I have NO idea how one might prove that.
...
Your result is absolute correct!(Clapping)
Hint: [sp] $(1-x^2)^m(1+x)^{2m}=(1-x)^m(1+x)^{3m}$[/sp]

Good luck! (Sun)
 
My solution
We start from the formula $(1-x^2)^m(1+x)^{2m}=(1-x)^m(1+x)^{3m}$
Now applying the binomial theorem
We have: $$ \sum_{k=0}^m (-1)^k{m\choose k}x^{2k} \sum_{j=0}^{2m} {2m\choose j}x^j = \sum_{k=0}^m (-1)^k{m\choose k}x^k \sum_{j=0}^{3m} {3m\choose j}x^j$$

\begin{equation} \label{eq1}\tag{1}\Leftrightarrow \sum_{k=0}^m \sum_{j=0}^{2m}(-1)^k {m\choose k} {2m\choose j}x^{2k+j} = \sum_{k=0}^m \sum_{j=0}^{3m} (-1)^k {m\choose k}{3m\choose j}x^{k+j} \end{equation}
compare coefficient of $x^{2m}$ in $\eqref{eq1} $ then we get
$\displaystyle \quad\sum_{2k+j=2m}(-1)^k {m\choose k} {2m\choose j}=\sum_{k+j=2m}(-1)^k {m\choose k} {3m\choose j}$
$\displaystyle \Leftrightarrow \sum_{k=0}^m(-1)^k {m\choose k} {2m\choose 2m-2k}=\sum_{k=0}^m (-1)^k {m\choose k} {3m\choose 2m-k}$
$\displaystyle \Leftrightarrow \sum_{k=0}^m(-1)^k {m\choose k} {2m\choose 2k}=\sum_{k=0}^m (-1)^k {m\choose k} {3m\choose m+k}$

with $m=2n$ then
\begin{equation} \label{eq2} \begin{aligned} S&=\sum_{k=0}^{2n}(-1)^k {2n\choose k} {4n\choose 2k}=\sum_{k=0}^{2n} (-1)^k {2n\choose k} {6n\choose 2n+k} \\ &=\sum_{k=0}^{2n} \dfrac{(-1)^k(2n)!(6n!)}{k!(2n-k)!(2n+k)!(4n-k)!} \\ &=\dfrac{(2n)!(6n)!}{(4n)!(4n)!} \sum_{k=0}^{2n} \dfrac{(-1)^k(4n)!(4n!)}{k!(4n-k)!(2n+k)!(2n-k)!} \\ &= \dfrac{(2n)!(6n)!}{(4n)!(4n)!} \sum_{k=0}^{2n} (-1)^k {4n\choose k} {4n\choose 2n-k} \\ &= \dfrac{(2n)!(6n)!}{(4n)!(4n)!} \sum_{k+j=2n} (-1)^k {4n\choose k} {4n\choose j}\tag{2} \end{aligned} \end{equation}
next to formula $(1-x^2)^{4r}=(1-x)^{4r}(1+x)^{4r}$
we get $ \displaystyle \sum_{k=0}^{4r}(-1)^k {4r\choose k}x^{2k}=\sum_{k=0}^{4r} \sum_{j=0}^{4r} (-1)^k {4r\choose k} {4r\choose j}x^{k+j}$
now, compare coefficient of $x^{2r}$ in that then we get
\begin{equation} \label{eq3}\tag{3} (-1)^r {4r\choose r}=\sum_{k+j=2r}(-1)^k {4r\choose k} {4r\choose j}\end{equation}
From $\eqref{eq2}$ and $\eqref{eq3}$, we have:
$$S=\dfrac{(2n)!(6n)!}{(4n)!(4n)!}\cdot(-1)^n{4n\choose n}=\dfrac{(-1)^n(2n)!(6n)!}{n!(3n)!(4n)!}$$
 
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