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 A167865 Number of partitions of n into distinct parts greater than 1, with each part divisible by the next. 23
 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 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 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: A303428 A223853 A023645 * A218654 A054571 A126865 Adjacent sequences:  A167862 A167863 A167864 * A167866 A167867 A167868 KEYWORD nonn,look,changed AUTHOR Max Alekseyev, Nov 13 2009 STATUS approved

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Last modified February 17 00:49 EST 2020. Contains 331976 sequences. (Running on oeis4.)