A317881 Number of series-reduced free pure identity multifunctions (with empty expressions allowed) with one atom and n positions.

1, 1, 1, 1, 3, 7, 15, 37, 91, 231, 593, 1557, 4111, 10941, 29295, 79087, 215015, 587463, 1611985, 4441473, 12284513, 34095797, 94931525, 265061363, 742029431, 2082310665, 5856540305, 16505796865, 46608877763, 131850193107, 373612733107, 1060339387939, 3013758348317

Offset: 1

Author

Gus Wiseman, Aug 09 2018

Keywords

• nonn

Comments

A series-reduced series-reduced free pure identity multifunction (with empty expressions allowed) (SRIM) 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 SRIM, k is an integer greater than or equal to 0 but not equal to 1, each of the g_i for i = 1, ..., k >= 0 is an SRIM, and for i != j we have g_i != g_j. The number of positions in an SRIM is the number of brackets [...] plus the number of o's.

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

Examples

			The a(6) = 7 SRIMs:
  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. A001003, A001678, A067824, A167865, A277996, A280000, A317875.

Cf. A317876, A317877, A317878, A317879, A317880.

Programs

Mathematica

allIdExprSR[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdExprSR[h],Select[Tuples[allIdExprSR/@p],UnsameQ@@#&]}],{p,If[g==0,{{}},Join@@Permutations/@Rest[IntegerPartitions[g]]]}]]];
Table[Length[allIdExprSR[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[k]))); v} \\ Andrew Howroyd, Sep 01 2018

Extensions

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