# Inspired by Mehul Chaturvedi

Algebra Level 5

$P(x)=\dfrac{ax^7}{b}+\dfrac{cx^6}{d}+\dfrac{ex^5}{f}+\dfrac{gx^4}{h}+\dfrac{ix^3}{j}+\dfrac{kx^2}{l}+\dfrac{mx}{n}-22$

Let a polynomial $$P(x)$$ be defined such that $$P(n)$$ is the number of open-chain structural isomers of $$n$$ carbon membered alkanes for $$n\in [1,8]$$.

Given that $$a$$, $$c$$, $$e$$, $$g$$, $$i$$,$$k$$ and $$m$$ are positive integers such that $$\gcd(a,b)=$$ $$\gcd(c,d)=$$ $$\gcd(e,f)=$$ $$\gcd(g,h)=$$ $$\gcd(i,j)=$$ $$\gcd(k,l)=$$ $$\gcd(m,n)= 1$$.

Calculate $$a+b+c+d+e+f+g+h+i+j+k+l+m+n-22$$.

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