0

So I'm working on a combinatorics problem, to which I've reached that $$f(1)=1, f(n)=1+\frac{1}{n}\sum_{k=1}^{n-1} f(k)$$ I have been working at this problem for a few days now, and I am fairly confident solving this recursive equation is the key to the problem. I tried somehow centering the recursive equation somehow, so that I would be able to solve it, since this is basically the only technique that I have learnt, but this seems completely unusuable for this equation. I notice that the equation diverges, so I am not asking for an asymptotic behavior, but more-so a closed form expression for $f(n)$. I know I haven't shown much of my progress, but thats because I really barely have any when it comes to solving this equation (when it comes to the original problem I guess I have the progress of reaching this equation, which I am confident is correct) Any help would be greatly greatly appreciated.

jbg05
  • 47
  • 4
    Check this: https://math.stackexchange.com/a/1318079/42969, or this: https://math.stackexchange.com/a/2594167/42969 – Martin R Mar 04 '23 at 10:34
  • 1
    Did you mean $\displaystyle\sum_{k=\color{red}0}^{n-1}f(k)$? – José Carlos Santos Mar 04 '23 at 10:40
  • I will look into these links. I have edited the question as per Jose Carlos Santos's comment. – jbg05 Mar 04 '23 at 10:54
  • Hello Anne and Martin, thanks for the links! I was unaware this would be looking at the Harmonic numbers... I thought the answer was supposed to be using Sterling numbers! I guess I misunderstood. Thanks anyways! – jbg05 Mar 04 '23 at 10:57

0 Answers0