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A120803 Number of series-reduced balanced trees with n leaves.

%I A120803 #14 Oct 26 2018 00:51:31
%S A120803 1,1,1,2,2,4,4,8,9,16,20,37,47,80,111,183,256,413,591,940,1373,2159,
%T A120803 3214,5067,7649,12054,18488,29203,45237,71566,111658,176710,276870,
%U A120803 437820,687354,1085577,1705080,2688285,4221333,6644088,10425748
%N A120803 Number of series-reduced balanced trees with n leaves.
%C A120803 In other words, rooted trees with all leaves at the same level and no node having exactly one child; the order of children is not significant.
%H A120803 Alois P. Heinz, <a href="/A120803/b120803.txt">Table of n, a(n) for n = 1..1000</a>
%F A120803 Let s_0(n) = 1 if n = 1, 0 otherwise; s_{k+1}(n) = EULER(s_k)(n) - s_k(n), where EULER is the Euler transform. Then a_n = sum_k s_k(n). (s_k(n) is the number of such trees of height k.) Note that s_k(n) = 0 for n < 2^k.
%e A120803 From _Gus Wiseman_, Oct 07 2018: (Start)
%e A120803 The a(10) = 16 series-reduced balanced rooted trees:
%e A120803   (oooooooooo)
%e A120803   ((ooooo)(ooooo))
%e A120803   ((oooo)(oooooo))
%e A120803   ((ooo)(ooooooo))
%e A120803   ((oo)(oooooooo))
%e A120803   ((ooo)(ooo)(oooo))
%e A120803   ((oo)(oooo)(oooo))
%e A120803   ((oo)(ooo)(ooooo))
%e A120803   ((oo)(oo)(oooooo))
%e A120803   ((oo)(oo)(ooo)(ooo))
%e A120803   ((oo)(oo)(oo)(oooo))
%e A120803   ((oo)(oo)(oo)(oo)(oo))
%e A120803   (((oo)(ooo))((oo)(ooo)))
%e A120803   (((oo)(oo))((ooo)(ooo)))
%e A120803   (((oo)(oo))((oo)(oooo)))
%e A120803   (((oo)(oo))((oo)(oo)(oo)))
%e A120803 (End)
%o A120803 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A120803 seq(n)={my(u=vector(n), v=vector(n)); u[1]=1; while(u, v+=u; u=EulerT(u)-u); v} \\ _Andrew Howroyd_, Oct 26 2018
%Y A120803 Cf. A000081, A000669, A001003, A001678, A007059, A048816, A079500, A119262, A244925, A316624, A320154, A320160, A320169, A320179.
%K A120803 nonn
%O A120803 1,4
%A A120803 _Franklin T. Adams-Watters_, Aug 18 2006