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%I #34 May 17 2021 14:59:28
%S 1,1,1,2,4,8,14,26,47,87,160,295,540,997,1832,3369,6197,11406,20975,
%T 38594,70991,130610,240275,442043,813184,1496053,2752251,5063319,
%U 9314879,17136632,31526032,57998423,106699160,196294065,361120800,664352454,1222204958
%N Number of series-reduced achiral free pure multifunctions (with empty expressions allowed) with one atom and n positions.
%C A series-reduced achiral expression (SRAE) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty but not unitary expression of the form h[g, ..., g], where h and g are SRAEs. The number of positions in an SRAE is the number of brackets [...] plus the number of o's.
%C Also the number of series-reduced achiral Mathematica expressions with one atom and n positions.
%H Andrew Howroyd, <a href="/A317884/b317884.txt">Table of n, a(n) for n = 1..500</a>
%F a(1) = 1; a(n > 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(6) = 8 SRAEs:
%e o[o,o,o,o]
%e o[o[],o[]]
%e o[][o,o,o]
%e o[][][o,o]
%e o[o,o,o][]
%e o[][o,o][]
%e o[o,o][][]
%e o[][][][][]
%p a:= proc(n) option remember; `if`(n=1, 1, a(n-1)+add(a(j)*add(
%p a(d), d=numtheory[divisors](n-j-1) minus {n-j-1}), j=1..n-1))
%p end:
%p seq(a(n), n=1..45); # _Alois P. Heinz_, Sep 05 2018
%t allAchExprSR[n_] := If[n == 1, {"o"}, Join @@ Cases[Table[PR[k, n - k - 1], {k, n - 1}], PR[h_, g_] :> Join @@ Table[Apply @@@ Tuples[{allAchExprSR[h], Select[Tuples[allAchExprSR /@ p], SameQ @@ # &]}], {p, If[g == 0, {{}}, Join @@ Permutations /@ Rest[IntegerPartitions[g]]]}]]]; Table[Length[allAchExprSR[n]], {n, 12}]
%t (* Second program: *)
%t a[n_] := a[n] = If[n == 1, 1, a[n-1] + Sum[a[j]*DivisorSum[
%t n-j-1, If[# < n-j-1, a[#], 0]&], {j, 1, n-2}]];
%t Array[a, 45] (* _Jean-François Alcover_, May 17 2021, after _Alois P. Heinz_ *)
%o (PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*(1 + 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]=v[n-1] + 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. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.
%Y Cf. A317882, A317883, A317885.
%K nonn
%O 1,4
%A _Gus Wiseman_, Aug 09 2018
%E Terms a(13) and beyond from _Andrew Howroyd_, Aug 19 2018
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