2
$\begingroup$

I have the following sequences:$$\begin{array} & & s_1(n)= & & & & & n\\ & s_2(n)= & & & & \dfrac {n^2}2+ &\dfrac n2\\ & s_3(n)= & & &\dfrac {n^3}3+&\dfrac {n^2}2+&\dfrac n6\\ & s_4(n)= & & \dfrac {n^4}4+ & \dfrac {n^3}2+ & \dfrac {n^2}4\\ & s_5(n)= & \dfrac {n^5}5+ & \dfrac {n^4}2+ & \dfrac {n^3}3- & & \dfrac n{30}\\ & \cdots & & \cdots & &\cdots & &\end{array}$$

And I'm trying to find the corresponding polynomial for this.

Questions:

  1. What is the polynomial that generates the coefficients of the above sequence?
  2. What are the steps?

Just simply by observing, I have the first two terms of $s_k(n)$ as$$s_k(n)=\dfrac {n^k}k+\dfrac {n^{k-1}}2+\text{the other terms that I don't know}\tag1$$ But the only problem is that $s_1(n)=n+\dfrac 12$. Which isn't right. Can you guys help me and complete $(1)$ while also improving what I already have?

  • 1
    It looks to me like you are looking at the formulas for sum of kth powers. – 2017-02-14
  • 1
    Try $s_k(n) = \frac{1}{k}\left[(n+1)^k-1 - \sum_{j=1}^{k-1} {k\choose j-1} s_{j}(n)\right]$. For more info see http://mathworld.wolfram.com/PowerSum.html or https://en.wikipedia.org/wiki/Faulhaber%27s_formula – 2017-02-14

0 Answers 0