What is highest coefficient ofthe polynomial $t(t+1)(t+2) \cdots (t+n)$ for arbitrary $n$?

Question

How to find the coefficient of in the polynomial

$t(t+1)(t+2) \cdots (t+n)$ for arbitrary $n$

?

According to me it should be the sum of the products of the numbers from $1,2,\cdots,n$ taking $k$ at a time i.e. it is a sum of $n \choose k$ products. Now if this is true then how do I find out the highest coefficient in that expansion?I feel real trouble in that case.I take quite a few examples and observe that the coefficient of $t^2$ is the highest among all other if $n \geq 2$.But I can't prove it in general.

Please help me in proving this.

Thank you in advance.

Up to a sign, these are the Stirling numbers of the first kind — Wojowu, May 24 '17 at 14:41
For $7 \le n \le 23$, the coefficient for $t^3$ is largest. At $n = 24$, the coefficient for $t^4$ takes over. not sure when will $t^5$ takes over. — achille hui, May 28 '17 at 07:37
we would need to know an (asymptotic) interpolating function for the Stirling 1 numbers in order to answer to your question. However it is sure that the max. is drifting to $\infty$ at the increasing of $n$. — G Cab, Jun 06 '17 at 11:06

Brethlosze · Answer 1 · 2017-05-24T15:17:22.587

1

Perhaps a missing index in here, but roughly this is:

$$ a_{k}=t^k \sum_{i=1}^{(n \ n+1-k)}\prod_{j \text{ in } C_i(n,n+1-k)} j $$

$C(n,k)$ is the set of all combinations of the integers from 1 to $n$, taking $k$ at each time, and $C_i$ is its $i$th element. The size of $C(n,k)$ is the binomial coefficient $(n \ k)=\frac{n!}{k!(n-k)!}$

$$ C(n,0)=\{\}\\ C(n,1)=\{\{1\},...,\{n\}\}\\ C(n,2)=\{\{1,2\},\{1,3\},...,\{n-2,n\},\{n-1,n\}\}\\ ...\\ C(n,n-1)=\{\{2,...n\},\{1,3,...,n\},\{1,...,n-2,n\},\{1,...,n-1\}\}\\ C(n,n)=\{\{1,...n\}\}\\ C(n,n+1)=\{\}\\ $$

edited May 24 '17 at 15:17

answered May 24 '17 at 14:51

Brethlosze

3,010

What would be the highest coefficient in that expansion? – Arnab Chattopadhyay. May 24 '17 at 16:04
$t^{n+1}+t^{n}\sum_{i=1}^ni+...+t\prod_{j=1}^nj$ – Brethlosze May 24 '17 at 17:53
Yeah I know that. But can you say that which coefficient is the largest?I guess it is the coefficent of $t^2$?Can you make it clear to me?I just draw this conclusion by taking quite a few examples.But that doesn't guarantee that the generalised result is true.Please help me in understanding it much clearly. – Arnab Chattopadhyay. May 25 '17 at 05:02
Can I use the fact that: $ \left[{n+1\atop k}\right] = n \left[{n\atop k}\right] + \left[{n\atop k-1}\right]$? – Arnab Chattopadhyay. May 25 '17 at 05:32
If you have a binomial coefficient, yes. But i dont see any....... – Brethlosze May 25 '17 at 14:22

score 1 · Answer 2 · answered Jun 08 '17 at 22:21

So we have that $$ t\left( {t + 1} \right) \cdots \left( {t + n} \right) = t^{\,\overline {\,n + 1\,} } $$ and $$ t^{\,\overline {\,n + 1\,} } = \sum\limits_{\,\left( {1\, \leqslant } \right)\,k\,\left( { \leqslant \,n + 1} \right)} {\left[ \begin{gathered} n + 1 \\ k \\ \end{gathered} \right]t^{\,k} } $$

Now $$ P(k;\,q) = {1 \over {q!}}\left[ \matrix{ q \cr k \cr} \right] $$ is a discrete Probability distribution
(see for instance this post) whose raw first and second moment are $$ \eqalign{ & E\left[ k \right] = \sum\limits_{\left( {0\, \le } \right)\,g\,\left( { \le \,q} \right)} {{k \over {q!}}\left[ \matrix{ q \cr k \cr} \right]} = H(q) = \gamma + \psi (q + 1) \cr & E\left[ {k^{\,2} } \right] = \sum\limits_{\left( {0\, \le } \right)\,k} {{{k^{\,2} } \over {q!}}\left[ \matrix{ q \cr k \cr} \right] = } \left( {H_{\,q} ^2 + H_{\,q} - {{\pi ^2 } \over 6} + \psi ^{\left( 1 \right)} \left( {q + 1} \right)} \right) \cr} $$

Your question concerns the mode of $\left[ \matrix{ q \cr k \cr} \right]$ which coincides with that of $P(k;\,q)$, but I could not find sufficient information on line about this distribution and its characteristics.

Plotting some examples of $P(k;\,q)$ it comes out that it has quite a peaked and symmetric shape concentrated in the lower values of $k$ and that the mode remains very close to the mean.
So ${\rm mode}\,P(k;q) \approx H(q)$.

If you need a more analytical result, then you can try and interpolate $P(k;\,q)$ by a continuous function, which has a closed expression for the mode.

What is highest coefficient ofthe polynomial $t(t+1)(t+2) \cdots (t+n)$ for arbitrary $n$?

2 Answers2