First-time poster here. While doing some research on Waring's problem and the term $\{(3/2)^n\},$ I determined that the following recurrence relation holds for a certain sequence (here $n$ is a fixed, positive integer):
$$r_{k}={n \choose k}+\frac{1}{2}r_{k-1}, r_0=1;\,0\leq k\leq n$$
Mathematica gave me the result $r_k = 3^n/2^k-2 {n\choose k+1} {}_{2}F_1(1,k-n+1;k+2;-2),$ which I was able to use with fruitful results, but I have no idea how to derive it by hand (here ${}_2F_1$ is the hypergeometric function). If someone could show me, I would be grateful. I'd also be willing to post a rough draft of my paper if someone was interested. Thanks!