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A345233 G.f. A(x) satisfies: A(x) = x + x^2 / exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

%I A345233 #4 Jun 11 2021 18:26:05
%S A345233 1,1,-1,0,1,0,-2,1,3,-4,-3,11,-2,-22,21,32,-72,-18,180,-95,-350,496,
%T A345233 449,-1542,125,3638,-3161,-6393,12780,5636,-35993,14509,77907,-97880,
%U A345233 -116880,337924,24514,-869531,631306,1692540,-2949009,-1933940,9035577,-2312868,-21166895
%N A345233 G.f. A(x) satisfies: A(x) = x + x^2 / exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).
%F A345233 G.f.: x + x^2 / Product_{n>=1} (1 + x^n)^a(n).
%F A345233 a(n+2) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * a(d) ) * a(n-k+2).
%t A345233 nmax = 45; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[(-1)^k A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%t A345233 a[1] = a[2] = 1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 45}]
%Y A345233 Cf. A007560, A049075, A345232.
%K A345233 sign
%O A345233 1,7
%A A345233 _Ilya Gutkovskiy_, Jun 11 2021