A281146 -id:A281146 - cp's OEIS frontend

A289078 Number of orderless same-trees of weight n.

1, 2, 2, 5, 2, 9, 2, 22, 6, 11, 2, 94, 2, 13, 12, 334, 2, 205, 2, 210, 14, 17, 2, 7218, 8, 19, 68, 443, 2, 1687, 2, 69109, 18, 23, 16, 167873, 2, 25, 20, 89969, 2, 7041, 2, 1548, 644, 29, 2, 36094795, 10, 3078, 24, 2604, 2, 1484102, 20, 1287306, 26, 35, 2

Offset: 1

Gus Wiseman, Jun 23 2017

nonn

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.

			The a(6)=9 orderless same-trees are: 6, (33), (3(111)), (222), (22(11)), (2(11)(11)), ((11)(11)(11)), ((111)(111)), (111111).

Alois P. Heinz, Table of n, a(n) for n = 1..2500

Cf. A196545, A273873, A275870, A281145, A281146, A289079.

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

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

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

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

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

Original entry on oeis.org

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

Gus Wiseman, Jun 23 2017

nonn

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

			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.

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

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

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

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

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

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

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

A305610 a(n) = (-1)^(n-1) + Sum_{d|n, d>1} binomial(a(n/d) + d - 1, d).

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 2, 0, 6, 3, 2, 11, 2, 3, 12, 0, 2, 38, 2, 11, 14, 3, 2, 90, 8, 3, 68, 11, 2, 127, 2, 0, 18, 3, 16, 1194, 2, 3, 20, 90, 2, 173, 2, 11, 644, 3, 2, 5158, 10, 68, 24, 11, 2, 12762, 20, 90, 26, 3, 2, 12910, 2, 3, 1386, 0, 22, 289, 2, 11, 30, 219, 2

Offset: 1

Gus Wiseman, Jun 06 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A063834, A196545, A220418, A273866, A273873, A275870, A281145, A281146, A289078, A289079, A289501, A290261, A290971, A291441, A300862-A300866.

a[n_]:=a[n]=(-1)^(n-1)+Sum[Binomial[a[n/d]+d-1,d],{d,Divisors[n]//Rest}];
Array[a,40]

A305610(n) = ((-1)^(n-1) + sumdiv(n,d,if(d==1,0,binomial(A305610(n/d)+d-1, d)))); \\ Antti Karttunen, Dec 05 2021

cp's OEIS Frontend

A289078 Number of orderless same-trees of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A305610 a(n) = (-1)^(n-1) + Sum_{d|n, d>1} binomial(a(n/d) + d - 1, d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI