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%I #13 Oct 28 2020 10:27:25
%S 1,1,2,3,5,6,12,13,24,33,60,61,142,143,289,447,699,700,1558,1559,3518,
%T 5375,8977,8978,17179,20305,40471,54808,98182,98183,242068,242069,
%U 477002,695051,1183654,1510612,2629806,2629807,5057173,7928654,12366025,12366026
%N Number of ordered rooted trees where each subtree from given node has the same number of nodes.
%H Alois P. Heinz, <a href="/A127525/b127525.txt">Table of n, a(n) for n = 1..5590</a>
%F a(1) = 1; a(n+1) = Sum_{d|n} a(n/d)^d.
%F L.g.f.: -log(Product_{n>=1} (1 - a(n)*x^n)^(1/n)) = Sum_{n>=1} a(n+1)*x^n/n. - _Ilya Gutkovskiy_, Apr 29 2019
%e The tree shown below left counts, because the left subtree has 3 nodes and so does the right subtree and a similar condition holds for the subtrees. The tree shown on the right is not counted, because the left subtree has 3 nodes, while the right subtree has 4.
%e O..........O...O...O
%e |..........|....\./.
%e O...O...O..O.....O..
%e .\...\./....\....|..
%e .O...O......O...O..
%e ..\./........\./...
%e ...O..........O....
%p a:= proc(n) option remember; `if`(n<2, n, add(
%p a((n-1)/d)^d, d=numtheory[divisors](n-1)))
%p end:
%p seq(a(n), n=1..45); # _Alois P. Heinz_, Sep 08 2018
%t a[1] = 1;
%t a[n_] := a[n] = Sum[a[(n-1)/d]^d, {d, Divisors[n-1]}];
%t Array[a, 45] (* _Jean-François Alcover_, Oct 28 2020 *)
%Y Cf. A000108, A127524, A007059.
%K nonn
%O 1,3
%A _Franklin T. Adams-Watters_, Jan 17 2007
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