A303022 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and no unitary parts (subexpressions of the form x[y]).

1, 1, 1, 2, 5, 12, 27, 63, 152, 376, 939, 2371, 6047, 15577, 40429, 105637, 277625, 733518, 1947126, 5190503, 13888811, 37291968, 100444019, 271316998, 734802247, 1994873116, 5427893149, 14799525982, 40429761365, 110645688034, 303316712450, 832799212777

Offset: 1

Author

Gus Wiseman, Aug 15 2018

Keywords

• nonn

Comments

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

Examples

			The a(6) = 12 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[][][][][]
		

Links

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

Crossrefs

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

Cf. A303023, A303024, A303025, A303026, A303027.

Programs

Mathematica

allOLBF[n_]:=allOLBF[n]=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allOLBF[h],Select[Union[Sort/@Tuples[allOLBF/@p]],Length[#]!=1&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allOLBF[n]],{n,10}]

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, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018