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%I #7 May 30 2023 19:38:39
%S 1,1,1,3,6,17,42,120,330,962,2797,8334,24989,75905,232142,715830,
%T 2220473,6928411,21723883,68424327,216376757,686742855,2186771571,
%U 6984248840,22368127861,71818903891,231132440916,745454242656,2409080380316,7799945417349
%N G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / (k*x^k) ).
%t nmax = 30; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[A[x^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
%t a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]
%o (PARI) seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p,x)-1); p = x + x^2*exp(sum(k=1, m, subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p + O(x*x^n)) \\ _Andrew Howroyd_, May 30 2023
%Y Cf. A005750, A007562, A363388.
%K nonn
%O 1,4
%A _Ilya Gutkovskiy_, May 30 2023