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%I #11 May 03 2019 21:23:47
%S 1,0,1,1,3,4,10,17,37,70,150,299,634,1311,2786,5879,12584,26904,58005,
%T 125242,271819,591297,1290976,2825170,6199964,13635749,30057649,
%U 66386206,146903289,325637240,723024160,1607805207,3580476340,7984266625,17827226469
%N Number of free pure achiral multifunctions with one atom and n positions.
%C A free pure achiral multifunction (PAM) 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 PAMs. The number of positions in a PAM is the number of brackets [...] plus the number of o's.
%H Andrew Howroyd, <a href="/A317883/b317883.txt">Table of n, a(n) for n = 1..200</a>
%F a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).
%F G.f. A(x) satisfies: A(x) = x * (1 + A(x) * Sum_{k>=1} A(x^k)). - _Ilya Gutkovskiy_, May 03 2019
%e The a(7) = 10 PAMs:
%e o[o[o[o]]]
%e o[o[o][o]]
%e o[o][o[o]]
%e o[o[o]][o]
%e o[o][o][o]
%e o[o[o,o,o]]
%e o[o][o,o,o]
%e o[o,o][o,o]
%e o[o,o,o][o]
%e o[o,o,o,o,o]
%t a[n_]:=If[n==1,1,Sum[a[k]*Sum[a[d],{d,Divisors[n-k-1]}],{k,n-2}]];
%t Array[a,12]
%o (PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*sum(k=1, n-2, 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-2, v[i]*sumdiv(n-i-1, d, v[d]))); v} \\ _Andrew Howroyd_, Aug 19 2018
%Y Cf. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.
%Y Cf. A317882, A317884, A317885.
%K nonn
%O 1,5
%A _Gus Wiseman_, Aug 09 2018
%E Terms a(13) and beyond from _Andrew Howroyd_, Aug 19 2018
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