I have computed the following terms in a sequence:
$\left\{1,-\frac{\pi ^2}{72},\frac{\pi ^4}{21600},-\frac{\pi ^6}{1270080},\frac{\pi ^8}{36288000},-\frac{\pi ^{10}}{632282112},\dots\right\}$
I am trying to guess what sequence they form. I suspect it has to do with Bernoulli numbers, as when I divide by the sequence
$-(-1)^nB_{2n-2}(2n)!^{-1}\pi^{2n-2}$
for $n=1,2,\dots$
then I get the simpler sequence
$\left\{2,2,-1,\frac{4}{3},-3,10\right\}$
But I still cannot guess the full sequence... I have tried looking on online lists of sequences with no luck.