A317876 Number of free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions.

1, 1, 2, 4, 10, 25, 67, 184, 519, 1489, 4342, 12812, 38207, 114934, 348397, 1063050, 3262588, 10064645, 31190985, 97061431, 303165207, 950115502, 2986817742, 9415920424, 29760442192, 94286758293, 299377379027, 952521579944, 3036380284111, 9696325863803

Offset: 1

Author

Gus Wiseman, Aug 09 2018

Keywords

• nonn

Comments

A free pure symmetric identity multifunction (with empty expressions allowed) (FOI) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where h is an FOI, each of the g_i for i = 1, ..., k >= 0 is an FOI, and for i < j we have g_i < g_j under a canonical total ordering such as the Mathematica ordering of expressions. The number of positions in an FOI is the number of brackets [...] plus the number of o's.

Also the number of free orderless identity Mathematica expressions with one atom and n positions.

Examples

			The a(5) = 10 FOIs:
  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. A000081, A004111, A052893, A053492, A277996, A280000, A317652, A317653, A317654, A317875.

Cf. A317877, A317878, A317879, A317880, A317881.

Programs

Mathematica

allIdExpr[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdExpr[h],Select[Union[Sort/@Tuples[allIdExpr/@p]],UnsameQ@@#&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allIdExpr[n]],{n,12}]

PARI

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

Formula

From Ilya Gutkovskiy, Apr 30 2019: (Start)

G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} (-1)^(k+1)*A(x^k)/k)).

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + (Sum_{n>=1} a(n)*x^n) * Product_{n>=1} (1 + x^n)^a(n)). (End)

Extensions

Terms a(16) and beyond from Andrew Howroyd, Aug 19 2018