Inspired by Mehul Chaturvedi

Chemistry Level 4

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

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 $$\gcd(a,b), \gcd(c,d) , \gcd(e,f) , \gcd(g,h) ,\gcd(i,j) , \gcd(k,l) , \gcd(m,n)$$ are all equal to 1.

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

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