A317884 Number of series-reduced achiral free pure multifunctions (with empty expressions allowed) with one atom and n positions.

1, 1, 1, 2, 4, 8, 14, 26, 47, 87, 160, 295, 540, 997, 1832, 3369, 6197, 11406, 20975, 38594, 70991, 130610, 240275, 442043, 813184, 1496053, 2752251, 5063319, 9314879, 17136632, 31526032, 57998423, 106699160, 196294065, 361120800, 664352454, 1222204958

Offset: 1

Author

Gus Wiseman, Aug 09 2018

Keywords

• nonn

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.

Examples

			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[][][][][]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.

Cf. A317882, A317883, A317885.

Programs

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 *)

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(dAndrew Howroyd, Aug 19 2018

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).

Extensions

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018