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%I #6 Jun 03 2023 14:22:13
%S 1,1,1,4,10,35,113,405,1447,5369,20143,76908,296800,1157784,4554142,
%T 18050308,72003513,288880549,1164867528,4718481975,19190711729,
%U 78338352168,320851617424,1318115448886,5430133003281,22427330328214,92847100210382,385217596191075,1601483701650310
%N G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^3 / (k*x^(2*k)) ).
%t nmax = 29; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[A[x^k]^3/(k x^(2 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[a[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d + 2], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 29}]
%Y Cf. A007562, A052751, A363387, A363466, A363467.
%K nonn
%O 1,4
%A _Ilya Gutkovskiy_, Jun 03 2023