OFFSET
1,3
COMMENTS
An unlabeled rooted tree is uniform if the multiplicities of the branches directly under any given node are all equal.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..2250 (terms 1..200 from Andrew Howroyd)
Gus Wiseman, All 72 uniform rooted trees with 8 nodes.
FORMULA
a(n) ~ c * d^n / n^(3/2), where d = 2.774067238136373782458114960391469140405537808253... and c = 0.43338208953061974806801546569720246018271214... - Vaclav Kotesovec, Sep 07 2019
EXAMPLE
The a(5) = 8 uniform rooted trees:
((((o))))
(((oo)))
((o(o)))
((ooo))
(o((o)))
(o(oo))
((o)(o))
(oooo)
MATHEMATICA
purt[n_]:=Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]], SameQ@@Length/@Split[#]&], {ptn, IntegerPartitions[n-1]}];
Table[Length[purt[n]], {n, 10}]
PROG
(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)); v=concat(v, sumdiv(n-1, d, t[d]))); v} \\ Andrew Howroyd, Aug 28 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 05 2018
EXTENSIONS
Term a(21) and beyond from Andrew Howroyd, Aug 28 2018
STATUS
approved