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%I #5 May 09 2019 22:03:54
%S 1,1,0,-2,4,0,-24,68,20,-936,3800,-2172,-64120,417752,-959852,
%T -5464092,68816400,-328509112,8361828,13473157664,-119068537700,
%U 448312242012,1868795480588,-43889516937276,348963546501928,-1006637409183472,-12316833304447344,217304253286437480
%N G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 + x)) * (A(x/(1 + x)) + A(x^2/(1 + x)^2) + A(x^3/(1 + x)^3) + ...)).
%F a(1) = 1; a(n+1) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k) * Sum_{d|k} a(d).
%t terms = 28; A[_] = 0; Do[A[x_] = x (1 + 1/(1 + x) Sum[A[x^k/(1 + x)^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
%t a[n_] := a[n] = Sum[(-1)^(n - k - 1) Binomial[n - 1, k] Sum[a[d], {d, Divisors[k]}], {k, 1, n - 1}]; a[1] = 1; Table[a[n], {n, 1, 28}]
%Y Cf. A003238, A014619, A308023.
%K sign
%O 1,4
%A _Ilya Gutkovskiy_, May 09 2019