deletions | additions       diff --git a/untitled.tex b/untitled.tex  index c8d736c..d4dd812 100644  --- a/untitled.tex  +++ b/untitled.tex 
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a_{k-5} = -\frac{2^{k-13}}{15}(9864(k-5)+3030(k-5)^2+515(k-5)^3+30(k-5)^4+(k-5)^5) \\  a_{k-6} = \frac{2\cdot2^{n-17}}{45}(125640(k-6)+90634(k-6)^2+12915(k-6)^3+1165(k-6)^4+45(k-6)^5+(k-6)^6)  \end{equation}  The we can see that the powers of 2 follow a sequence $0,1,3,4,7,8,10,11,15,16$ which is potentially A005187, the number of ones in the binary expansion of $2n$, and the denominators of the convergents of $1/\sqrt{1-x}$. $1/\sqrt{1-x}$, we will denote this quantity $\xi(n)$, with $\xi(0)=0,\xi(1)=1,\cdots$.  It then appears that for $a_{k-m}$ the coefficent of the highest power of $(k-m)$ is $1$, the coefficent of the next highest power is $3m(m+1)/2=3,9,18,30,35,63...$. We also see the sequence $1,1,1,3,3,15,45,315,315,2835$ is A049606, the largest odd divisor of $n!$. This is very useful. We can attempt to factor $n!$ out of the coefficients.  Then, letting $\chi(n)$ be A011371, the number of binary digits in $n$, we have \begin{equation}  a_k = \frac{2^{\chi(0)}}{0!}2^{k-\xi(0)}\\  a_{k-1} = -\frac{2^{\chi(1)}{1!}(k)2^{k-\xi(1)} \\  a_{k-2} = \frac{2^\chi(2)}{2!}(3k+k^2)2^{k-\xi(2)} \\  a_{k-3} = -\frac{2^\chi(3)}{3!}\frac{2^{k-\xi(3)}}{1}(38k+9k^2+k^3) \\  a_{k-4} = \frac{2^{k-7}}{3}(378k+179k^2+18k^3+k^4) \\  a_{k-5} = -\frac{2^{k-8}}{15}(9864k+3030k^2+515k^3+30k^4+k^5) \\  a_{k-6} = \frac{2\cdot2^{k-11}}{45}(125640k+90634k^2+12915k^3+1165k^4+45k^5+k^6)\\  a_{k-7} = \frac{2\cdot2^{k-12}}{315}(1684080k+2003652k^2+463204k^3+40005k^4+2275k^5+63k^6+k^7)\\  a_{k-8} = \frac{2^{k-15}}{315}(42089040k+50017932k^2+14438676k^3+1728769k^4+101640k^5+4018k^6+84k^7+k^8)\\  a_{k-9} = \frac{2^{k-16}}{2835}(415900800k+1431527472k^2+492751916k^3+69663132k^4+5253969k^5+225288k^6+6594k^7+108k^8+k^9)\\  \end{equation}