deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 2b7eb4b..726a3dd 100644  --- a/untitled.tex  +++ b/untitled.tex 
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and  \begin{equation}  a_k = \kappa(0)\\  a_{k-1} = \kappa(1) \kappa(1)k  \\ a_{k-2} = \kappa(2)(3k+k^2) \\  a_{k-3} = \kappa(3)(38k+9k^2+k^3) \\  a_{k-4} = \kappa(4)(378k+179k^2+18k^3+k^4) \\ 
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a_{k-7} = \kappa(7)(1684080k+2003652k^2+463204k^3+40005k^4+2275k^5+63k^6+k^7)\\  a_{k-8} = \kappa(8)(42089040k+50017932k^2+14438676k^3+1728769k^4+101640k^5+4018k^6+84k^7+k^8)\\  a_{k-9} = \kappa(9)(415900800k+1431527472k^2+492751916k^3+69663132k^4+5253969k^5+225288k^6+6594k^7+108k^8+k^9)\\  \end{equation} Now we focus on the polynomial aspect in $k$. We may define a function $\pi_n(k)$ such that \begin{equation}  a_{k-m}= \kappa(m)\pi_m(k)  \end{equation}  then we have \begin{equation}  \pi_0(k)=1\\  \pi_1(k)=k\\  \pi_2(k)=3k+k^2\\  \pi_3(k)=38k+9k^2+k^3\\  \pi_4(k)=378k+179k^2+18k^3+k^4  \end{equation}