A196545 Number of weakly ordered plane trees with n leaves.

1, 1, 2, 5, 12, 34, 92, 277, 806, 2500, 7578, 24198, 75370, 243800, 776494, 2545777, 8223352, 27221690, 88984144, 296856400, 979829772, 3287985078, 10934749788, 36912408342, 123519937044, 418650924886, 1408867195252, 4794243983204, 16205061000480

Offset: 1

Author

Gus Wiseman, Oct 03 2011

Keywords

• nonn

Comments

A weakly ordered plane tree (p-tree) of weight n > 1 is a sequence (t_1, t_2, ..., t_k) where k > 1 and for some integer partition y of length k and sum n, the term t_i is a p-tree of weight y_i, for 1 <= i <= k. For n = 1, the only p-tree is a single node.

This definition precludes nodes with only one branch, and non-leaf nodes have weight 0. If the above is changed so that k >= 1 and y is partition of n-1, we get the trees counted by A093637. Binary p-trees are counted by A000992.

Examples

			Let o denote a single node. The 12 p-trees of weight 5 are: ((((oo)o)o)o), (((ooo)o)o), (((oo)(oo))o), (((oo)oo)o), ((oooo)o), (((oo)o)(oo)), ((ooo)(oo)), (((oo)o)oo), ((ooo)oo), ((oo)(oo)o), ((oo)ooo), (ooooo).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• David Callan, The weakly ordered plane trees of size up to 4
• Gus Wiseman, All 277 weakly ordered plane trees of weight 8

Crossrefs

Cf. A000041, A063834, A093637, A000992.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(i>n, 0, a(i)*`if`(i=n, 1, b(n-i, i)))+`if`(i>1, b(n, i-1), 0)
    end:
a:= proc(n) option remember;
      `if`(n=1, 1, add(a(k)*b(n-k, min(n-k, k)), k=1..n-1))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 06 2012

Mathematica

PTNum[n_] := PTNum[n] =
    If[n === 1, 1, Plus @@ Function[y,
    Times @@ PTNum /@ y] /@ Rest[Partitions[n]]]; Array[PTNum, 20]
(* Second program: *)
b[n_, i_] := b[n, i] = If[i>n, 0, a[i]*If[i == n, 1, b[n-i, i]]] + If[i>1, b[n, i-1], 0];
a[n_] := a[n] = If[n == 1, 1, Sum[a[k]*b[n-k, Min[n-k, k] ], {k, 1, n-1}]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

PARI

seq(n)={my(v=vector(n)); v[1] = 1; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); v} \\ Andrew Howroyd, Aug 26 2018

Sage

@cached_function
def A196545(n):
    if n == 1: return 1
    return sum(prod(A196545(t) for t in p) for p in Partitions(n,min_length=2))
# D. S. McNeil, Oct 03 2011

Formula

a(1) = 1; for n > 1, a(n) = Sum_{y} Product_{i} a(y_i) where the sum is over all partitions of n with at least two parts.

The generating function is characterized by the formal equation 1 + 2*S(x) = x + 1/P(x) where S(x) = Sum_{n>0} a(n)*x^n and P(x) = Product_{n>0} (1 - a(n)*x^n) are the formal infinite series and formal infinite product with a(n) as coefficients and roots respectively.