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Number of rooted plane trees where every branch that has a predecessor (a branch directly to its left and emanating from the same root) has at least as many leaves as its predecessor.
6

%I #11 Jan 22 2021 15:43:25

%S 1,1,2,5,13,34,90,242,660,1822,5085,14333,40759,116817,337140,979098,

%T 2859439,8393113,24747052,73262246,217681621,648939319,1940461444,

%U 5818595438,17492367097,52712114792,159193762250,481754196170,1460650624068,4436422703787,13496947320929

%N Number of rooted plane trees where every branch that has a predecessor (a branch directly to its left and emanating from the same root) has at least as many leaves as its predecessor.

%H Andrew Howroyd, <a href="/A304173/b304173.txt">Table of n, a(n) for n = 1..200</a>

%F G.f.: A(x,1) where A(x,y) satisfies A(x,y) = x*(y-1 + 1/(Product_{k>=1} 1 - y^k * [y^k] A(x,y))). - _Andrew Howroyd_, Jan 22 2021

%e The a(5) = 13 plane trees:

%e ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),

%e (((o))o), (o((o))), (o(oo)), ((o)(o)),

%e ((o)oo), (o(o)o), (oo(o)),

%e (oooo).

%e Missing from this list is ((oo)o).

%t pplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[pplane/@c],OrderedQ[Count[#,{},{0,Infinity}]&/@#]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];

%t Table[Length[pplane[n]],{n,10}]

%o (PARI) seq(n)={my(p=x*y+O(x^2)); for(n=2, n, p=x*(y-1 + 1/prod(k=1, n-1, 1 - y^k*polcoef(p,k,y)))); Vec(subst(p,y,1))} \\ _Andrew Howroyd_, Jan 22 2021

%Y Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304175.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 16 2018

%E Terms a(15) and beyond from _Andrew Howroyd_, Jan 22 2021