OFFSET
1,4
COMMENTS
A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
FORMULA
E.g.f. A(x) satisfies A(x) = x + A(exp(x)-x-1). - Ira M. Gessel, Nov 17 2021
EXAMPLE
The a(1) = 1 through a(5) = 11 rooted trees:
1 (12) (123) (1234) (12345)
((12)(34)) ((12)(345))
((13)(24)) ((13)(245))
((14)(23)) ((14)(235))
((15)(234))
((23)(145))
((24)(135))
((25)(134))
((34)(125))
((35)(124))
((45)(123))
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
phy2[labs_]:=If[Length[labs]==1, labs, Union@@Table[Sort/@Tuples[phy2/@ptn], {ptn, Select[sps[Sort[labs]], Length[#1]>1&]}]];
Table[Length[Select[phy2[Range[n]], SameQ@@Length/@Position[#, _Integer]&]], {n, 7}]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n, k)={my(u=vector(n), v=vector(n)); u[1]=k; while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n, k, b(n, k)))); vector(n, k, sum(i=1, k, binomial(k, i)*(-1)^(k-i)*M[i, k]))} \\ Andrew Howroyd, Oct 26 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 06 2018
EXTENSIONS
Terms a(10) and beyond from Andrew Howroyd, Oct 26 2018
STATUS
approved