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A345884 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} (-1)^k * A(x^k) / k).

%I A345884 #6 Jun 28 2021 15:55:12
%S A345884 1,-2,7,-26,103,-442,1982,-9122,42985,-206526,1007322,-4974066,
%T A345884 24819268,-124949782,633882799,-3237261340,16629986395,-85873762466,
%U A345884 445491479309,-2320717519612,12134813554225,-63667883444468,335083404759136,-1768545061282712,9358571746569760
%N A345884 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} (-1)^k * A(x^k) / k).
%F A345884 G.f.: x / Product_{n>=1} (1 + x^n)^(2*a(n)).
%F A345884 a(n+1) = (2/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * a(d) ) * a(n-k+1).
%p A345884 a:= proc(n) option remember; `if`(n=1, 1, 2*add(a(n-k)*add(d*a(d)
%p A345884        *(-1)^(k/d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
%p A345884     end:
%p A345884 seq(a(n), n=1..25);  # _Alois P. Heinz_, Jun 28 2021
%t A345884 nmax = 25; A[_] = 0; Do[A[x_] = x Exp[2 Sum[(-1)^k A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%t A345884 a[1] = 1; a[n_] := a[n] = (2/(n - 1)) Sum[Sum[(-1)^(k/d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 25}]
%Y A345884 Cf. A000151, A005753, A049075, A345885.
%K A345884 sign
%O A345884 1,2
%A A345884 _Ilya Gutkovskiy_, Jun 28 2021