A317877 Number of free pure identity multifunctions with one atom and n positions.

1, 0, 1, 0, 2, 2, 5, 10, 18, 46, 94, 212, 476, 1058, 2441, 5564, 12880, 29920, 69620, 163220, 383376, 904114, 2139592, 5074784, 12074152, 28789112, 68803148, 164779064, 395373108, 950416330, 2288438591, 5518864858, 13329183894, 32237132814, 78069124640

Offset: 1

Author

Gus Wiseman, Aug 09 2018

Keywords

• nonn

Comments

A free pure identity multifunction (PIM) is either (case 1) the leaf symbol "o", or (case 2) an expression of the form h[g_1, ..., g_k] where h is a PIM, each of the g_i for i = 1, ..., k > 0 is a PIM, and for i != j we have g_i != g_j. The number of positions in a PIM is the number of brackets [...] plus the number of o's.

Examples

			The a(8) = 10 PIMs:
  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]
		

Links

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

Crossrefs

Cf. A000081, A001003, A003238, A004111, A052893, A053492, A277996, A280000, A317875.

Cf. A317876, A317878, A317879, A317880, A317881.

Programs

Mathematica

allIdPMF[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-2}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdPMF[h],Select[Tuples[allIdPMF/@p],UnsameQ@@#&]}],{p,Join@@Permutations/@IntegerPartitions[g]}]]];
Table[Length[allIdPMF[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]=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