A303027 Number of free pure symmetric multifunctions with one atom, n positions, and no empty or unitary parts (subexpressions of the form x[] or x[y]).

1, 0, 0, 1, 1, 1, 3, 5, 7, 15, 28, 47, 90, 175, 319, 607, 1181, 2251, 4325, 8449, 16425, 31992, 62823, 123521, 243047, 480316, 951290, 1886293, 3749341, 7467815, 14893500, 29752398, 59532947, 119274491, 239275400, 480638121, 966571853, 1945901716, 3921699524

Offset

1,7

Comments

Also the number of orderless Mathematica expressions with one atom, n positions, and no empty or unitary parts.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Example

The a(10) = 15 Mathematica expressions:
  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[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,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,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][o,o]
  o[o,o,o,o,o,o,o,o]

Mathematica

allOLZR[n_]:=allOLZR[n]=If[n==1, {"o"}, Join@@Cases[Table[PR[k, n-k-1], {k, n-1}], PR[h_, g_]:>Join@@Table[Apply@@@Tuples[{allOLZR[h], Select[Union[Sort/@Tuples[allOLZR/@p]], Length[#]>1&]}], {p, IntegerPartitions[g]}]]];
Table[Length[allOLZR[n]], {n, 25}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)-v); v=concat(v, sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Crossrefs

Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.

Cf. A303022, A303023, A303024, A303025, A303026.

Sequence in context: A165785 A362057 A261646 * A217615 A334084 A050553

Adjacent sequences: A303024 A303025 A303026 * A303028 A303029 A303030

Keyword

nonn

Author

Gus Wiseman, Aug 15 2018

Extensions

Terms a(29) and beyond from Andrew Howroyd, Aug 19 2018

Status

approved