OFFSET
1,2
COMMENTS
An orderless same-tree t is either: (case 1) a positive integer, or (case 2) a finite multiset of two or more orderless same-trees, all having the same weight. The weight of t in case 1 is the number itself, and in case 2 it is the sum of weights of the branches. For example {{{3,{1,1,1}},{2,{1,1},{1,1}}},{{{1,1,1},{1,1,1}},{{1,1},{1,1},{1,1}}}} is an orderless same-tree of weight 24 with 2 branches.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..2500
FORMULA
a(n) = 1 + Sum_{d|n, d>1} binomial(a(n/d)+d-1, d).
EXAMPLE
The a(6)=9 orderless same-trees are: 6, (33), (3(111)), (222), (22(11)), (2(11)(11)), ((11)(11)(11)), ((111)(111)), (111111).
MAPLE
with(numtheory):
a:= proc(n) option remember; 1 + add(
binomial(a(n/d)+d-1, d), d=divisors(n) minus {1})
end:
seq(a(n), n=1..60); # Alois P. Heinz, Jul 05 2017
MATHEMATICA
a[n_]:=If[n===1, 1, 1+Sum[Binomial[a[n/d]+d-1, d], {d, Rest[Divisors[n]]}]];
Array[a, 100]
PROG
(PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sumdiv(n, d, binomial(v[n/d]+d-1, d))); v} \\ Andrew Howroyd, Aug 20 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 23 2017
STATUS
approved