<s>[INST] What is the theory behind a simple pattern involving binomial coefficients? These binomial equations are all true.
$\binom{7}{4} = \binom{4}{4} + 3[ \binom{4}{3} +  \binom{4}{2}] +  \binom{4}{1}$
$\binom{8}{4} = \binom{5}{4} + 3[ \binom{5}{3} +  \binom{5}{2}] +  \binom{5}{1}$
$\binom{9}{4} = \binom{6}{4} + 3[ \binom{6}{3} +  \binom{6}{2}] +  \binom{6}{1}$
$\dots$

Is this the result of some general binomial identity or theory?

My work
I stumbled upon this while working with a recurrence relation in $(m,n)$ with $m,n \in \Bbb Z^+$
that I was trying to put into closed form.
 [/INST] By repeatedly applying Pascal's Theorem:
$$\begin{align*}
\binom{n}{4} &=\binom{n-1}{4}+\binom{n-1}{3}
 \\ &=\left(\binom{n-2}{4}+\binom{n-2}{3}\right)+\left(\binom{n-2}{3}+\binom{n-2}{2}\right)
 \\ &=\left(\left(\binom{n-3}{4}+\binom{n-3}{3}\right)+\left(\binom{n-3}{3}+\binom{n-3}{2}\right)\right)+\left(\left(\binom{n-3}{3}+\binom{n-3}{2}\right)+\left(\binom{n-3}{2}+\binom{n-3}{1}\right)\right)
\\ &= \binom{n-3}{4}+3\binom{n-3}{3}+3\binom{n-3}{2}+\binom{n-3}{1}
\end{align*}
$$
</s>