A052893 Number of objects generated by the Combstruct grammar defined in the Maple program. See the link for the grammar specification.

1, 1, 3, 10, 37, 144, 589, 2483, 10746, 47420, 212668, 966324, 4439540, 20587286, 96237484, 453012296, 2145478716, 10215922013, 48877938369, 234862013473, 1132902329028, 5483947191651, 26630419098206, 129696204701807, 633339363924611, 3100369991303297

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Number of free pure symmetric multifunctions with n + 1 unlabeled leaves. A free pure symmetric multifunction f in PSM is either (case 1) f = the leaf symbol "o", or (case 2) f = an expression of the form h[g_1, ..., g_k] where k > 0, h is in PSM, each of the g_i for i = 1, ..., k is in PSM, and for i < j we have g_i <= g_j under a canonical total ordering of PSM, such as the Mathematica ordering of expressions. - Gus Wiseman, Aug 02 2018

Examples

			From _Gus Wiseman_, Aug 02 2018: (Start)
The a(3) = 10 free pure symmetric multifunctions with 4 unlabeled leaves:
  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]
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 869
• Maplesoft, Combstruct grammars.
• Mathematica Reference, Orderless.

Crossrefs

Cf. A001003, A052891, A277996, A279944, A280000.

Programs

Maple

spec := [S, {C = Set(B,1 <= card), B=Prod(Z,S), S=Sequence(C)}, unlabeled]:
seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
a[n_]:=a[n]=If[n==1,1,Sum[a[k]*Sum[Product[multing[a[First[s]],Length[s]],{s,Split[p]}],{p,IntegerPartitions[n-k]}],{k,1,n-1}]];
Array[a,30] (* Gus Wiseman, Aug 02 2018 *)

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=1, n, v=Vec(1/(1-x*Ser(EulerT(v))))); v} \\ Andrew Howroyd, Aug 09 2020

Formula

G.f.: 1/(1 - g(x)) where g(x) is the g.f. of A052891. - Andrew Howroyd, Aug 09 2020

Extensions

More terms from Gus Wiseman, Aug 02 2018