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I wonder if it is possible to find a closed form expression for following sequence: \begin{equation*} C_1=1 \end{equation*} \begin{equation*} C_2=x^2+\frac 32 \end{equation*} \begin{equation*} C_3=x^4+5x^2+2 \end{equation*} \begin{equation*} C_4=x^6+11x^4+\frac {115}{8}x^2+\frac 58 \end{equation*} \begin{equation*} C_5=x^8+20x^6+56x^4+32x^2+3 \end{equation*} Is there any method using Wolfram Mathematica? These are coefficients in perturbative solution of a following equation on $\tau$: \begin{equation*} \frac{1}{y^4}\tau^4-\frac{2}{y^2}\tau^3-\left(\frac 1{y^4} +x^2-1\right)\tau^2+\frac{2}{y^2}\tau-1=0 \end{equation*} were $x>1$ and $y\gg1$

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    $\begingroup$ How did you generate the sequence? $\endgroup$
    – Frank Vel
    Feb 26, 2015 at 20:47

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