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%I #8 Aug 19 2018 13:22:44
%S 1,0,0,1,1,1,2,3,4,7,9,14,21,32,45,69,103,153,224,338,500,746,1107,
%T 1645,2447,3652,5413,8052,11993,17834,26500,39447,58655,87240,129772,
%U 193001,287034,427014,635048,944501,1404910,2089633,3107864,4622670,6875533
%N Number of series-reduced free pure achiral multifunctions with one atom and n positions.
%C A series-reduced free pure achiral multifunction (SRAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty and non-unitary expression of the form h[g, ..., g] where h and g are SRAMs. The number of positions in a SRAM is the number of brackets [...] plus the number of o's.
%H Andrew Howroyd, <a href="/A317885/b317885.txt">Table of n, a(n) for n = 1..500</a>
%F a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1), d < n - k - 1} a(d).
%e The a(10) = 7 SRAMs:
%e o[o[o,o],o[o,o]]
%e o[o,o][o,o][o,o]
%e o[o,o][o,o,o,o,o]
%e o[o,o,o][o,o,o,o]
%e o[o,o,o,o][o,o,o]
%e o[o,o,o,o,o][o,o]
%e o[o,o,o,o,o,o,o,o]
%t a[n_]:=If[n==1,1,Sum[a[k]*Sum[a[d],{d,Most[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=2, 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, if(d<n-i-1, v[d], 0)))); v} \\ _Andrew Howroyd_, Aug 19 2018
%Y Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A317853, A317875.
%Y Cf. A317882, A317883, A317884.
%K nonn
%O 1,7
%A _Gus Wiseman_, Aug 09 2018
%E Terms a(17) and beyond from _Andrew Howroyd_, Aug 19 2018
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