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I'm trying to find an asymptotic expansion for the following sum:

$$\sum_{k=2}^{n}\frac{n!}{k\left(n-k\right)!}=\sum_{k=2}^{n}\left(\begin{array}{c} n\\ k \end{array}\right)\left(k-1\right)!$$

for large $n$. According to Maple, this sum is equivalent to:

$$\frac{n^{2}-n}{2}\,_{3}F_{1}\left(1,2,2-n;3;-1\right)$$

where $_{3}F_{1}$ is a special case of the generalized hypergeometric function. But on the internet I cannot find any asymptotic expansion of $_{3}F_{1}$, because most work focused on other cases, e.g. $_{2}F_{1}$. I think it could be easier to work directly on the original sum. But by searching in previous posts I cannot find the specific sum I'm interested in. Moreover, asymptotic approximations of $\left(\begin{array}{c} n\\ k \end{array}\right)$ do not look very useful since they do not work for $k\approx n$. Help would be very appreciated!

user2983638
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3 Answers3

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Can be done in a similar way to this. Let $k = n - k$ for convenience: $$S = \sum_{k = 2}^n \frac {n!} {k (n - k)!} = \sum_{k = 0}^{n - 2} \frac {n!} {(n - k) k!} = (n - 1)! \sum_{k = 0}^{n - 2} \frac 1 {k!} \frac 1 {1 - \frac k n}.$$ The main contribution comes from small values of $k$, and it is sufficient to use an approximation of the summand that holds for large $n$ and fixed $k$. The $i$th order term in the approximation will be simply $(k/n)^i$. Then the summation range in $S$ can be extended to infinity, and we get a complete asymptotic series for $S$: $$S \approx e (n - 1)! \sum_{i = 0}^\infty \frac {B_i} {n^i},$$ where $B_0 = 1, B_1 = 1, .\!.\!.$ are the Bell numbers.

Maxim
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  • Oh that's great! Now I see what you've done. So, just to explain better your derivation, first you used the Taylor expansion of $\frac{1}{1-x}$, where in our case $x=\frac{k}{n}$. This provides the terms $\left(\frac{k}{n}\right)^{i}$. Then you used the Dobinski's formula. In this way you got the Bell numbers. Very cool! Just a question: can we really say that $\sum_{k=0}^{n-2}\frac{k^{i}}{k!}\approx eB_{i}$ for every $i$, even though $n$ is finite? I would expect that for large $i$ this identity is not true anymore. – user2983638 May 11 '18 at 13:03
  • But clearly this is an asymptotic expansion, so $n$ should be large anyway, which probably answers my question. – user2983638 May 11 '18 at 13:09
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    I actually used $\approx$ in the asymptotic series sense: $$ \forall i_0 \geqslant 0 \enspace \frac S {e (n - 1)!} - \sum_{i = 0}^{i_0} \frac {B_i} {n^i} = o(n^{-i_0}), \quad n \to \infty.$$ It can be proved along the same lines as in the post I linked to. – Maxim May 11 '18 at 15:02
  • Thanks! I've also added an additional answer, where I show a numerical confirmation of your result. – user2983638 May 11 '18 at 16:10
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$$\sum_{k=2}^{n}\binom{n}{k}(k-1)! = \sum_{k=2}^{n}\binom{n}{k}\int_{0}^{+\infty}x^{k-1}e^{-x}\,dx = \int_{0}^{+\infty}\frac{(1+x)^n-nx-1}{x}e^{-x}\,dx $$ is expected to behave like $\int_{0}^{+\infty}x^{n-1}e^{-x}\,dx = (n-1)!$ plus a perturbation due to the fact that $\frac{(1+x)^n-nx-1}{x}$ does not behave like $x^{n-1}$ in a right neighbourhood of the origin. More accurate approximations can be derived from Laplace's method.

Jack D'Aurizio
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  • Wow absolutely brilliant! Very concise and clear explanation. Many thanks! – user2983638 Mar 09 '17 at 14:06
  • More precisely, the sum is $\Theta((n - 1)!)$, but it's not asymptotically equivalent to $(n - 1)!$. I've added an answer. – Maxim May 09 '18 at 16:16
  • @Maxim Could you explain what is the $\Theta$ function you are mentioning? I also ran numerical simulations to check the Jack D'Aurizio's answer, and I have to say it seems correct. – user2983638 May 10 '18 at 12:11
  • It's "grows as": $f = \Theta(g)$ means $f = O(g) \land g = O(f)$. Consider that the last term in the original sum is asymptotically equivalent to $(n - 1)!$, and the next to last term is also a.e. to $(n - 1)!$; their sum cannot be a.e. to $(n - 1)!$. – Maxim May 10 '18 at 15:14
  • Ok, actually I only verified numerically that the series grows as $\left(n-1\right)!$ and not as $n!$ or $\left(n-2\right)!$. This means that I missed the coefficient $e\sum_{i=0}^{\infty}\frac{B_{i}}{n^{i}}$. – user2983638 May 11 '18 at 13:18
  • I would say the coefficient $e$ :). This holds: $S \sim e (n - 1)!$. It's an asymptotic series though, the sum $\sum_{i = 0}^\infty B_i/n^i$ diverges for any $n$. – Maxim May 11 '18 at 15:12
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This is just a numerical confirmation of the result derived by Maxim. The figure below shows that Maxim's asymptotic series (red line, truncated to $i=10$ in this example) converges to the original series (blue line) for large $n$, and that both the series converge to Euler's number $e$. enter image description here

user2983638
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