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A052893
Number of objects generated by the Combstruct grammar defined in the Maple program. See the link for the grammar specification.
23
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
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
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from Gus Wiseman, Aug 02 2018
STATUS
approved