<s>[INST] Bernoulli Number analog using Cosine I know that Bernoulli Numbers can be found with the generating function
$$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$
I was wondering if any work has been done using a similar equation
$$\frac{x^2}{\cos{x}-1}=\sum_{n=0}^{\infty}\frac{C_n}{n!}x^{2n}$$
I'm particularly interested in the $C_n$.  Can anyone help me with a reference to work with this particular generating function or to the coefficients $C_n$?
 [/INST] Your sequence (negated) starts
$$
2, \frac{1}{6}, \frac{1}{60}, \frac{1}{504}, \frac{1}{3600}, \frac{1}{22176}, \frac{691}{82555200}, \frac{1}{570240}, \frac{3617}{8821612800}, \frac{43867}{414096883200}, \frac{174611}{5822264448000}, \ldots
$$
We know that
$$
\frac{1-\cos x}{x^2} = \frac{1}{2!} - \frac{1}{4!}x^2 + \frac{1}{6!}x^4 - \frac{1}{8!}x^6 + \cdots.
$$
Let $c_n = C_n/n!$. The formula for the inverse of a power series gives
$$
c_0 = 2, \quad c_n = 2\frac{c_{n-1}}{4!} - 2\frac{c_{n-2}}{6!} + 2\frac{c_{n-3}}{8!} - 2\frac{c_{n-4}}{10!} + \cdots.
$$
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