### ironman7453's blog

By ironman7453, history, 4 months ago, ,

I have been trying to solve this problem.

We need to find the ($N+1-K)^{th}$ coefficient of the polynomial $(x+N)(x+N+1)(x+N+2)...(x+M-1)(x+M)$.

When $K=3$, $C(N,M,K)$ can be easily calculated by the triple sum $\sum_{a=N}^{M}\sum_{b=a+1}^{M}\sum_{c=b+1}^{M}abc$

As $K$ becomes larger, finding closed form of the sum becomes difficult. I tried another approach. When $M-N$ and $K$ are fixed, $C(N,M,K)$ can be expressed as a polynomial of degree $K$ on the variable $N$.

For example, if $M-N=10$ and $K=3$, $C(N,N+10,3)=18150+11880N+2475N^2+165N^3$.

Final answer should be a polynomial of degree $50$. But to find the coefficients, I need $50$ data points. So far I have been unable to evaluate those. Any help will be much appreciated.

P.S. — This problem is not from an ongoing competition.

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 » 4 months ago, # |   0 Did you find the Solution?