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A319123 Number of series-reduced plane trees with n leaves such that each branch directly under any given node has a different number of leaves.

%I A319123 #5 Sep 12 2018 15:10:35
%S A319123 1,1,3,7,21,75,277,1083,4419,18493,77729,332557,1444477,6307225,
%T A319123 27912147,123878207,554733045,2492087531,11280537097,51120499279,
%U A319123 233319480419,1065835004917,4895443823281,22505853359485,103958158302085,480365303903637,2229412587062123
%N A319123 Number of series-reduced plane trees with n leaves such that each branch directly under any given node has a different number of leaves.
%e A319123 The a(4) = 7 plane trees:
%e A319123   (oooo)
%e A319123   (o(ooo))
%e A319123   ((ooo)o)
%e A319123   (o(o(oo)))
%e A319123   (o((oo)o))
%e A319123   ((o(oo))o)
%e A319123   (((oo)o)o)
%t A319123 b[n_]:=b[n]=1+Sum[Times@@b/@f,{f,Join@@Permutations/@Select[IntegerPartitions[n],And[Length[#]>1,UnsameQ@@#]&]}];
%t A319123 Array[b,30]
%Y A319123 Cf. A000108, A001003, A007853, A074206, A118376, A273873, A277130, A281113, A304173, A304175, A319122.
%K A319123 nonn
%O A319123 1,3
%A A319123 _Gus Wiseman_, Sep 11 2018