| MATHEMATICA
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a[0] := 1; a[n_?OddQ] := 0; a[n_?EvenQ] := a[n] = -Sum[a[k] Binomial[n, k], {k, 0, n - 1, 2}]; Map[a, Range[0; , 30]] (* _Oliver Seipel_, May 20 2024 *)
a[0] := 1; a[n_?OddQ] := 0; a[n_?EvenQ] :=] := a[n] = -Sum[a[n - k]/k!, {k, 2, n, 2}]; Map[#! a[#] &, Range[0, 30]] (* _Oliver Seipel_, May 20 2024 *)
a[n] = -Sum[a[k] Binomial[n, k], {k, 0, n - 1, 2}]; Map[a,
Range[0, 30]] (* Oliver Seipel, May 20 2024 *)
a[0] := 1; a[n_?OddQ] := 0;
a[n_?EvenQ] := a[n] = -Sum[a[n - k]/k!, {k, 2, n, 2}]; Map[#! a[#] &,
Range[0, 30]] (* Oliver Seipel, May 20 2024 *)
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