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%I #11 Apr 30 2019 21:50:19
%S 1,1,3,9,30,102,369,1362,5181,20064,79035,315366,1272789,5185080,
%T 21296196,88083993,366584253,1533953100,6449904138,27238006971,
%U 115475933202,491293053093,2096930378415,8976370298886,38528771056425,165784567505325
%N Number of achiral free pure multifunctions with n unlabeled leaves.
%C An achiral free pure multifunction is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g], where h and g are both achiral free pure multifunctions.
%H Andrew Howroyd, <a href="/A317875/b317875.txt">Table of n, a(n) for n = 1..200</a>
%F a(1) = 1; a(n > 1) = Sum_{0 < k < n} a(n - k) * Sum_{d|k} a(d).
%F From _Ilya Gutkovskiy_, Apr 30 2019: (Start)
%F G.f. A(x) satisfies: A(x) = x + A(x) * Sum_{k>=1} A(x^k).
%F G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x + (Sum_{n>=1} a(n)*x^n) * (Sum_{n>=1} a(n)*x^n/(1 - x^n)). (End)
%e The first 4 terms count the following multifunctions.
%e o,
%e o[o],
%e o[o,o], o[o[o]], o[o][o],
%e o[o,o,o], o[o[o][o]], o[o[o[o]]], o[o[o,o]], o[o][o,o], o[o][o[o]], o[o][o][o], o[o,o][o], o[o[o]][o].
%t a[n_]:=If[n==1,1,Sum[a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
%t Array[a,12]
%o (PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*(sum(k=1, n-1, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ _Andrew Howroyd_, Aug 19 2018
%o (PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-1, v[i]*sumdiv(n-i, d, v[d]))); v} \\ _Andrew Howroyd_, Aug 19 2018
%Y Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317853.
%Y Cf. A317876, A317877, A317878, A317879, A317880, A317881.
%Y Cf. A317882, A317883, A317884, A317885.
%K nonn
%O 1,3
%A _Gus Wiseman_, Aug 09 2018
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