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A363062 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 A363062 #7 May 18 2023 10:41:33
%S A363062 1,-1,-1,0,1,1,-1,-2,0,4,4,-5,-13,-2,26,30,-29,-94,-26,189,246,-198,
%T A363062 -769,-302,1512,2228,-1372,-6691,-3425,12672,21046,-9503,-60776,
%U A363062 -38353,109719,205330,-61001,-567518,-427145,967914,2045196,-314417,-5405209,-4743873,8625547
%N A363062 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 A363062 G.f.: x - x^2 / Product_{n>=1} (1 - x^n)^a(n).
%F A363062 a(1) = 1, a(2) = -1; a(n) = (1/(n - 2)) * Sum_{k=1..n-2} ( Sum_{d|k} d * a(d) ) * a(n-k).
%t A363062 nmax = 45; A[_] = 0; Do[A[x_] = x - x^2 Exp[Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%t A363062 a[1] = 1; a[2] = -1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 45}]
%Y A363062 Cf. A007562, A045648, A345235.
%K A363062 sign
%O A363062 1,8
%A A363062 _Ilya Gutkovskiy_, May 16 2023