%I #4 Jun 11 2021 21:14:59
%S 1,1,4,10,36,135,504,2000,8072,33099,138132,582930,2485412,10692219,
%T 46340984,202175344,887175352,3913032212,17338327848,77141235796,
%U 344491008296,1543591834950,6937783312048,31270131096820,141305878384704,640065923118435,2905664234243052,13217615913137250
%N G.f. A(x) satisfies: A(x) = x + x^2 * exp(4 * Sum_{k>=1} (-1)^(k+1) * A(x^k) / k).
%F G.f.: x + x^2 * Product_{n>=1} (1 + x^n)^(4*a(n)).
%F a(n+2) = (4/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k+2).
%t nmax = 28; A[_] = 0; Do[A[x_] = x + x^2 Exp[4 Sum[(-1)^(k + 1) A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%t a[1] = a[2] = 1; a[n_] := a[n] = (4/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 28}]
%Y Cf. A007560, A052772, A345243, A345244.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, Jun 11 2021