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%I #5 Jun 11 2021 21:15:36
%S 1,1,4,14,52,205,832,3492,14960,65322,289384,1298064,5882712,26897352,
%T 123919576,574718308,2681028168,12571650355,59222213028,280139215118,
%U 1330101884932,6336757979653,30282375754944,145124083402256,697293746743760,3358385599930269,16210842955175380
%N G.f. A(x) satisfies: A(x) = x + x^2 * exp(4 * Sum_{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} d * a(d) ) * a(n-k+2).
%t nmax = 27; A[_] = 0; Do[A[x_] = x + x^2 Exp[4 Sum[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[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 27}]
%Y Cf. A007562, A052763, A345200, A345241.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, Jun 11 2021