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%I #6 Jun 03 2023 14:22:16
%S 1,1,1,5,15,61,240,1019,4387,19462,87649,401077,1856698,8685295,
%T 40978465,194806667,932141498,4486014160,21699575863,105443142514,
%U 514469464550,2519437043753,12379461876092,61013509071216,301553269618318,1494229881209940,7421627743464582,36942997716584746
%N G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^4 / (k*x^(3*k)) ).
%t nmax = 28; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[A[x^k]^4/(k x^(3 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%t a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[f[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d + 3], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 28}]
%Y Cf. A007562, A052773, A363387, A363465, A363468.
%K nonn
%O 1,4
%A _Ilya Gutkovskiy_, Jun 03 2023