#JavaScript (ES6), 83 bytes

0-indexed.

<!-- language-all: lang-javascript -->

    f=(n,i=3,a=[p=1,1,1,1])=>a[n]||f(n,i+1,[...a,p+=(g=k=>k<i&&a[k]*a[i-++k]+g(k))(2)])

[Try it online!][TIO-k0qope9q]

[JavaScript (Node.js)]: https://nodejs.org
[TIO-k0qope9q]: https://tio.run/##HY1BCoMwEEX3PcUsRDJNDNUudbxICDhYFRtJREs36tlTLX/xFu/Bf/OX13YZ50/mw6uLsSfh1UhPxWRmytV/Fqlm4@2@95eVuTJaa1azJDGQo9pVY5qycfbOZsykdFYOwiGKAi3GPizCA8GjBA8VFBelRNhuAG3wa5g6PYVBNCySzR94psl2PuHRYHk74g8 "JavaScript (Node.js) – Try It Online"