A167865 Number of partitions of n into distinct parts greater than 1, with each part divisible by the next.

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 3, 3, 3, 1, 5, 1, 5, 4, 3, 1, 6, 2, 5, 4, 5, 1, 9, 1, 6, 4, 4, 4, 8, 1, 6, 6, 7, 1, 11, 1, 8, 8, 4, 1, 10, 3, 10, 5, 8, 1, 11, 4, 10, 7, 6, 1, 13, 1, 10, 11, 7, 6, 15, 1, 9, 5, 11, 1, 14, 1, 9, 12, 8, 5, 15, 1, 16, 9, 8, 1, 18, 5, 12, 7, 10, 1, 21, 7, 13, 11, 5

Offset

0,7

Comments

Number of lone-child-avoiding achiral rooted trees with n + 1 vertices, where a rooted tree is lone-child-avoiding if all terminal subtrees have at least two branches, and achiral if all branches directly under any given vertex are equal. The Matula-Goebel numbers of these trees are given by A331967. - Gus Wiseman, Feb 07 2020

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Formula

a(0) = 1 and for n>=1, a(n) = Sum_{d|n, d>1} a((n-d)/d).

G.f. A(x) satisfies: A(x) = 1 + x^2*A(x^2) + x^3*A(x^3) + x^4*A(x^4) + ... - Ilya Gutkovskiy, May 09 2019

Example

a(12) = 4: [12], [10,2], [9,3], [8,4].
a(14) = 3: [14], [12,2], [8,4,2].
a(18) = 5: [18], [16,2], [15,3], [12,6], [12,4,2].
From Gus Wiseman, Jul 13 2018: (Start)
The a(36) = 8 lone-child-avoiding achiral rooted trees with 37 vertices:
  (oooooooooooooooooooooooooooooooooooo)
  ((oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo))
  ((ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo))
  ((ooooo)(ooooo)(ooooo)(ooooo)(ooooo)(ooooo))
  ((oooooooo)(oooooooo)(oooooooo)(oooooooo))
  (((ooo)(ooo))((ooo)(ooo))((ooo)(ooo))((ooo)(ooo)))
  ((ooooooooooo)(ooooooooooo)(ooooooooooo))
  ((ooooooooooooooooo)(ooooooooooooooooo))
(End)

Maple

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(a((n-d)/d), d=divisors(n) minus{1}))
    end:
seq(a(n), n=0..200);  # Alois P. Heinz, Mar 28 2011

Mathematica

a[0] = 1; a[n_] := a[n] = DivisorSum[n, a[(n-#)/#]&, #>1&]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 07 2015 *)

Prog

(PARI) { A167865(n) = if(n==0, return(1)); sumdiv(n, d, if(d>1, A167865((n-d)\d) ) ) }

Crossrefs

Cf. A001678, A067824, A122651, A167439, A167865, A167866, A184998, A316782.

The semi-achiral version is A320268.

Matula-Goebel numbers of these trees are A331967.

The semi-lone-child-avoiding version is A331991.

Achiral rooted trees are counted by A003238.

Cf. A000081, A000669, A289079, A320222, A331912, A331933, A331936, A331992.

Sequence in context: A223853 A023645 A366769 * A218654 A054571 A336158

Adjacent sequences: A167862 A167863 A167864 * A167866 A167867 A167868

Keyword

nonn,look

Author

Max Alekseyev, Nov 13 2009

Status

approved