I came across an integer sequence for a Ramanujan-type formula for $1/\pi$, namely, $$a(n)=\color{brown}{\sum_{k=0}^n \binom{n}{k}\binom{2n}{n}\binom{2k}{k}^{-1}}\sum_{j=0}^k\binom{k}{j}^4 = 1, 4, 36, 424, 5716, 83568, 1289352,\dots$$
where $a(0)=1$. Q: Is there any way to simplify the brown part, maybe reduce the number of binomials, or even reduce $a(n)$ to a single-summation?
P.S. I tried to investigate the properties of the brown part, and it seems to be related to a number triangle given by A046521, but I can't get it to yield the numbers above.