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 *)
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Max Alekseyev, Nov 13 2009
STATUS
approved