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a(2n-1) = A000670(n), a(2n) = A005649(n). - Michael Somos, May 04 2022

G.f. = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 13*x^5 + 44*x^6 + 75*x^7 + ... - Michael Somos, May 04 2022

a[ n_] := If[n < 0, 0, Sum[ k!*(StirlingS2[Quotient[n+1, 2], k] + StirlingS2[Quotient[n+2, 2], k]), {k, 0, n+1}]/2]; (* Michael Somos, May 04 2022 *)

a[ n_] := If[n < 0, 0, With[{m = Quotient[n+1, 2]},

m!*SeriesCoefficient[1/(2 - Exp@x)^Mod[n, 2, 1], {x, 0, m}]]]; (* Michael Somos, May 04 2022 *)

Cf. A000670, A005649, A326895.

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Michael Somos: Added more info.

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Andrew Howroyd, <a href="/A327868/b327868.txt">Table of n, a(n) for n = 0..200</a>

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Case n = 4: The a(4) = 8 achiral loops are: