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Without using any special functions:
e@0=1;e@n_:=-Sum[Binomial[n,2k]e[nn!Sum[e[n-2k]k]/k!/(n-k)!,{k,2,n/,2}]
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e@0=1;e@n_:=-Sum[Binomial[n,2k]e[n-2k],{k,n/2}]
Try it online!
e@0=1;e@n_:=-n!Sum[e[n-k]/k!/(n-k)!,{k,2,n,2}]
