<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] Here is a more natural derivation of the formula:
$$
\frac{z^2}{1 - \cos z} =
2\sum_{n = 0}^\infty (-1)^{n - 1}\frac{(2n - 1)B_{2n}}{(2n)!}z^{2n}
\ \ \ (\left\lvert{z}\right\rvert < 2\pi).
$$
We start with the defining result that
for all $z \in \mathbb{C}$ such that $\left\lvert{z}\right\rvert < 2\pi$,
$$
\frac{z}{e^z - 1} =
\sum_{m = 0}^\infty \frac{B_m}{m!}z^m.
$$
Then:
\begin{gather*}
\coth\frac{z}{2} =
\frac{\cosh\frac{z}{2}}{\sinh\frac{z}{2}} =
\frac{e^z + 1}{e^z - 1}
= \frac{2}{e^z - 1} + 1, \\
\therefore\
\frac{z}{2}\coth\frac{z}{2} =
\frac{z}{e^z - 1} + \frac{z}{2}, \\
\therefore\
\frac{z}{2}\cot\frac{z}{2} =
\frac{iz}{2}\coth\frac{iz}{2} =
\frac{iz}{e^{iz} - 1} + \frac{iz}{2} \\
= \sum_{n = 0}^\infty (-1)^n\frac{B_{2n}}{(2n)!}z^{2n} +
iz\sum_{n = 0}^\infty (-1)^n\frac{B_{2n + 1}}{(2n + 1)!}z^{2n} +
\frac{iz}{2}
\end{gather*}
But this is a real-valued function of $z$ when $z$ is real,
therefore:
\begin{gather*}
B_1 = -\frac{1}{2}; \ \ \
B_{2n + 1} = 0 \ \ \ (n \geqslant 1); \\
\frac{z}{2}\cot\frac{z}{2} =
\sum_{n = 0}^\infty (-1)^n\frac{B_{2n}}{(2n)!}z^{2n}
\ \ \ (\left\lvert{z}\right\rvert < 2\pi).
\end{gather*}
(We could also have argued, earlier, that
$\frac{z}{2}\coth\frac{z}{2}$ is an even function of $z$.)
Differentiating the series term-by-term, and multiplying by $z$:
\begin{gather*}
\sum_{n = 0}^\infty (-1)^n\frac{2nB_{2n}}{(2n)!}z^{2n} =
z\frac{d}{dz}\left(\frac{z}{2}\cot\frac{z}{2}\right) \\
= \frac{z}{2}\cot\frac{z}{2} - \frac{z^2}{4}\mathrm{cosec}^2\frac{z}{2} 
= \frac{z}{2}\cot\frac{z}{2} - \frac{1}{2}\cdot\frac{z^2}{1 - \cos z},
\end{gather*}
which gives the required result.
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