A317879 Number of free pure identity multifunctions (with empty expressions allowed) with one atom and n positions.

1, 1, 2, 4, 11, 29, 83, 251, 767, 2403, 7652, 24758, 80875, 266803, 887330, 2972108, 10016981, 33942461, 115572864, 395226810, 1356840007, 4674552089, 16156355357, 56003840659, 194651585875, 678220460687, 2368505647624, 8288873657180, 29064904732911

Offset: 1

Author

Gus Wiseman, Aug 09 2018

Keywords

• nonn

Comments

A free pure identity multifunction (with empty expressions allowed) (IME) 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 IME, each of the g_i for i = 1, ..., k >= 0 is an IME, and for i != j we have g_i != g_j. The number of positions in an IME is the number of brackets [...] plus the number of o's.

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

Examples

			The a(5) = 11 IMEs:
  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. A000081, A001003, A004111, A277996, A280000, A317875.

Cf. A317876, A317877, A317878, 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[Tuples[allIdExpr/@p],UnsameQ@@#&]}],{p,Join@@Permutations/@IntegerPartitions[g]}]]];
Table[Length[allIdExpr[n]],{n,12}]

PARI

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, my(p=prod(k=1, n, 1 + sum(i=1, n\k, binomial(v[k], i)*x^(i*k)*y^i) + O(x*x^n))); v[n]=v[n-1]+sum(k=1, n-2, v[n-k-1]*subst(serlaplace(y^0*polcoef(p, k)), y, 1))); v} \\ Andrew Howroyd, Sep 01 2018

Extensions

Terms a(13) and beyond from Andrew Howroyd, Sep 01 2018