%I #6 Aug 13 2020 22:43:15

%S 1,-1,1,-2,4,-7,21,-51,113,-498,1088,-3335,21407,-14653,232389,

%T -1275288,-3636526,-44468245,-7468609,700603965,12178055777,

%U 67189448344,175549544778,-2432123216941,-36279392911507,-287078642854853,-945866835928323

%N E.g.f.: exp(1 + x^2/2 - exp(x)).

%F a(0) = 1; a(n) = -a(n-1) - Sum_{k=3..n} binomial(n-1,k-1) * a(n-k).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * (2*k-1)!! * A000587(n-2*k).

%t nmax = 26; CoefficientList[Series[Exp[1 + x^2/2 - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 26}]

%t Table[Sum[Binomial[n, 2 k] (2 k - 1)!! BellB[n - 2 k, -1], {k, 0, Floor[n/2]}], {n, 0, 26}]

%Y Cf. A000587, A001147, A097514, A293037, A293038.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Aug 13 2020