A304173 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.

1, 1, 2, 5, 13, 34, 90, 242, 660, 1822, 5085, 14333, 40759, 116817, 337140, 979098, 2859439, 8393113, 24747052, 73262246, 217681621, 648939319, 1940461444, 5818595438, 17492367097, 52712114792, 159193762250, 481754196170, 1460650624068, 4436422703787, 13496947320929

Offset: 1

Author

Gus Wiseman, Aug 16 2018

Keywords

• nonn

Examples

			The a(5) = 13 plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), (o(oo)), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list is ((oo)o).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

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

Programs

Mathematica

pplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[pplane/@c],OrderedQ[Count[#,{},{0,Infinity}]&/@#]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[pplane[n]],{n,10}]

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

Formula

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

Extensions

Terms a(15) and beyond from Andrew Howroyd, Jan 22 2021