The recurrence equivalent to the conjectural formula for rho sub 2,3

Let $c1_{2,3}(s,n)$ denote the quantity appearing in the preceding conjecture, with $s$ fixed and $n$ varying. The recurrence conjecture.

$$(n-1)\rho_{2,3}(s,n)=(n+2s-2)\rho_{2,3}(s,n-1).$$

The source explicitly says that, using the binomial identity shown immediately before this statement, proving the preceding formula is equivalent to proving this recurrence. No proof or resolution is supplied.