A289079 Number of orderless same-trees of weight n with all leaves equal to 1.

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 13, 1, 3, 3, 22, 1, 16, 1, 15, 3, 3, 1, 151, 2, 3, 6, 17, 1, 41, 1, 334, 3, 3, 3, 637, 1, 3, 3, 275, 1, 56, 1, 21, 19, 3, 1, 15591, 2, 27, 3, 23, 1, 902, 3, 516, 3, 3, 1, 7858, 1, 3, 21, 69109, 3, 98, 1, 27, 3, 67, 1, 811756, 1

Offset: 1

Author

Gus Wiseman, Jun 23 2017

Keywords

• nonn

Comments

a(n) is also the number of orderless same-trees of weight n with all leaves greater than 1.

Examples

			The a(12)=13 orderless same-trees with all leaves greater than 1 are: ((33)(33)), ((33)(222)), ((33)6), ((222)(222)), ((222)6), (66), ((22)(22)(22)), ((22)(22)4), ((22)44), (444), (3333), (222222), 12.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A196545, A273873, A275870, A281145, A281146, A289078.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1, add(
      binomial(a(n/d)+d-1, d), d=divisors(n) minus {1}))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jul 05 2017

Mathematica

a[n_]:=If[n===1,1,Sum[Binomial[a[n/d]+d-1,d],{d,Rest[Divisors[n]]}]];
Array[a,100]

PARI

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = sumdiv(n, d, binomial(v[n/d]+d-1, d))); v} \\ Andrew Howroyd, Aug 20 2018

Python

from sympy import divisors, binomial
l=[0, 1]
for n in range(2, 101): l+=[sum([binomial(l[n//d] + d - 1, d) for d in divisors(n)[1:]]), ]
l[1:] # Indranil Ghosh, Jul 06 2017

Formula

a(1) = 1, a(n>1) = Sum_{d|n, d>1} binomial(a(n/d)+d-1, d).