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

0,3

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

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.

Formula

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

Example

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A264231 A151053 A151054 * A052818 A299501 A226434

Adjacent sequences: A052890 A052891 A052892 * A052894 A052895 A052896

Keyword

easy,nonn

Author

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

Extensions

More terms from Gus Wiseman, Aug 02 2018

Status

approved