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

1,4

Comments

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

Links

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

Formula

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

Example

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.

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]

Prog

(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

Crossrefs

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

Sequence in context: A296121 A277120 A104725 * A249810 A257111 A011129

Adjacent sequences: A289076 A289077 A289078 * A289080 A289081 A289082

Keyword

nonn

Author

Gus Wiseman, Jun 23 2017

Status

approved