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%I #5 Jun 11 2021 21:15:12
%S 1,1,3,6,19,57,177,586,1950,6642,22990,80400,284346,1014237,3644841,
%T 13185810,47976382,175458798,644630064,2378084209,8805524949,
%U 32714828733,121917589291,455625246297,1707142362234,6411576477380,24133229559243,91023263056629,343964618949140,1302098673500514
%N G.f. A(x) satisfies: A(x) = x + x^2 * exp(3 * Sum_{k>=1} (-1)^(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} (-1)^(k/d+1) * d * a(d) ) * a(n-k+2).
%t nmax = 30; A[_] = 0; Do[A[x_] = x + x^2 Exp[3 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] = (3/(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, 30}]
%Y Cf. A007560, A052757, A345243, A345245.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, Jun 11 2021
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