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A167865 Number of partitions of n into distinct parts greater than 1, with each part divisible by the next. 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Number of series-reduced planted achiral trees with n nodes, where a rooted tree is series-reduced if all terminal subtrees have at least two branches, and achiral if all branches directly under any given node are equal. - Gus Wiseman, Jul 13 2018

LINKS

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

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(37) = 8 series-reduced planted achiral trees:

  (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-Fran├žois 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, A003238, A067824, A122651, A167439, A167865, A167866, A184998, A316782.

Sequence in context: A303428 A223853 A023645 * A218654 A054571 A126865

Adjacent sequences:  A167862 A167863 A167864 * A167866 A167867 A167868

KEYWORD

nonn,look

AUTHOR

Max Alekseyev, Nov 13 2009

STATUS

approved

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Last modified November 19 19:18 EST 2019. Contains 329323 sequences. (Running on oeis4.)