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Sep 6 at 12:14 comment added Peter Taylor Minor notes not worth bumping with an edit: it might be more useful to write the first D-finite recurrence as $\frac{B(n)_{r,1}}{n^k} = \frac{B(n-1)_{r,1}}{(n-1)^k} + (n-2)^k \frac{B(n-2)_{r,1}}{(n-k)^2}$. Since the degrees of the polynomials will be $ks$, if $k$ isn't very small relative to $n$ then the goal is probably to get $ks \sim n/s$, so take $s = \lfloor \sqrt{nk^{-1}} \rfloor$.
Sep 5 at 21:57 vote accept user369335
Sep 5 at 17:37 history edited Peter Taylor CC BY-SA 4.0
Missed a very important exponent
Sep 5 at 16:31 history answered Peter Taylor CC BY-SA 4.0