OFFSET
1,4
COMMENTS
A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
FORMULA
G.f.: A(x) satisfies A(x) = x + x*exp(Sum_{k>=1} A(x^k)/k) - x*exp(Sum_{k>=1} x^k*A(x^k)/(1 + x^k)/k). - Andrew Howroyd, Dec 10 2020
a(n) ~ c * d^n / n^(3/2), where d = 2.69474016697407303512228736537683134987637576... and c = 0.41800971384719166056172258174139385922545... - Vaclav Kotesovec, Nov 16 2021
EXAMPLE
The a(1) = 1 through a(6) = 11 trees:
o (o) (oo) (ooo) (oooo) (ooooo)
((oo)) ((ooo)) ((oooo))
(o(o)) (o(oo)) (o(ooo))
(oo(o)) (oo(oo))
((o(o))) (ooo(o))
((o)(oo))
((o(oo)))
((oo(o)))
(o((oo)))
(o(o)(o))
(o(o(o)))
MATHEMATICA
urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]], {ptn, IntegerPartitions[n-1]}];
Table[Length[Select[urt[n], FreeQ[#, q:{__List}/; Times@@Length/@q==1]&]], {n, 10}]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n)); v[1]=1; for(n=1, #v-1, v[n+1] = EulerT(v[1..n])[n] - EulerT(Vec(x^2*Ser(v[1..n-1])/(1+x), -n))[n]); v} \\ Andrew Howroyd, Dec 10 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2020
EXTENSIONS
Terms a(19) and beyond from Andrew Howroyd, Dec 10 2020
STATUS
approved