Finding a general expression for a recursively defined sequence.

Question

I have a sequence of numbers that I would like to find a general expression for. The sequence is defined as follows. $$C_m=\frac{1}{2m-3}\left[C_{m-1}+\frac{m^2-9m+16}{9\cdot 2^{m+4}(m-4)!}\right], m\geq 6$$ With the the seed being $C_5=-\frac{1}{5760}$.

The strategy would be to calculate several terms to try to find any patterns and then use induction to try to prove some general expression for $C_m$. I have calculated - by hand - the first 7 terms with no promising pattern in sight.

$$C_5=-\frac{1}{2^7\cdot3^2\cdot5}$$ $$C_6=-\frac{13}{2^{10}\cdot3^4\cdot5}$$ $$C_7=-\frac{1}{2^{11}\cdot3^4\cdot5}$$ $$C_8=\frac{1}{2^{12}\cdot3^4\cdot5}$$ $$C_9=\frac{1}{2^{10}\cdot3^5\cdot5^2}$$ $$C_{10}=\frac{19}{2^{17}\cdot3^5\cdot5^2}$$ $$C_{11}=\frac{29}{2^{18}\cdot3^5\cdot5^2\cdot7}$$

It it also tempting to use the recursive definition to try to get a sum representation for $C_m$, but this just gets ugly really quick.

I should also explain why I have reason to think that a general expression exists. The sequence is one sequence of a family of sequences that are in some sense connected to each other. All the previous sequences have had a provable general expression so thats why I suspect this would be possible for this sequence as well.

Does anyone see any patterns in the given terms or have any other strategies that could work?

Answer 1

One way is to use Mathematica. Upon giving Mathematica the code

RSolve[{a[n] == 
   1/(2 n - 3) (a[n - 1] + (n^2 - 9 n + 16)/(9*2^(n + 4)*(n - 4)!)), 
  a[5] == -1/5760}, a[n], n]

It outputs (in plain mathematical language) $$a_n=\frac{2^{-4-n}\left(5n^3-60n^2+199n-192\right)\Gamma\left(\frac{2n-1}{2}\right)}{405\sqrt{\pi}(n-3)!\prod_{k=1}^{n-1}\left(k-\frac{1}{2}\right)}$$