%I #6 Jun 11 2021 21:15:57

%S 1,1,3,9,28,93,315,1109,3969,14505,53726,201588,764001,2921730,

%T 11257881,43669590,170383933,668236581,2632898016,10416893159,

%U 41368099791,164841324837,658883345595,2641064296638,10613953319448,42757746556377,172628891937513,698398635475974

%N G.f. A(x) satisfies: A(x) = x + x^2 * exp(3 * Sum_{k>=1} A(x^k) / k).

%F G.f.: x + x^2 / Product_{n>=1} (1 - x^n)^(3*a(n)).

%F a(n+2) = (3/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).

%t nmax = 28; A[_] = 0; Do[A[x_] = x + x^2 Exp[3 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] = (3/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 28}]

%Y Cf. A006964, A007562, A345200, A345242.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Jun 11 2021