OFFSET
1,4
COMMENTS
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.
Also the number of series-reduced achiral Mathematica expressions with one atom and n positions.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
FORMULA
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).
EXAMPLE
The a(6) = 8 SRAEs:
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[][][][][]
MAPLE
a:= proc(n) option remember; `if`(n=1, 1, a(n-1)+add(a(j)*add(
a(d), d=numtheory[divisors](n-j-1) minus {n-j-1}), j=1..n-1))
end:
seq(a(n), n=1..45); # Alois P. Heinz, Sep 05 2018
MATHEMATICA
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}]
(* Second program: *)
a[n_] := a[n] = If[n == 1, 1, a[n-1] + Sum[a[j]*DivisorSum[
n-j-1, If[# < n-j-1, a[#], 0]&], {j, 1, n-2}]];
Array[a, 45] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018
STATUS
approved