A300443 Number of binary enriched p-trees of weight n.

1, 1, 2, 3, 8, 15, 41, 96, 288, 724, 2142, 5838, 17720, 49871, 151846, 440915, 1363821, 4019460, 12460721, 37374098, 116809752, 353904962, 1109745666, 3396806188, 10712261952, 33006706419, 104357272687, 323794643722, 1027723460639, 3204413808420, 10193485256501

Offset: 0

Author

Gus Wiseman, Mar 05 2018

Keywords

• nonn

Comments

A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.

Examples

			The a(4) = 8 binary enriched p-trees: 4, (31), (22), ((21)1), ((11)2), (2(11)), (((11)1)1), ((11)(11)).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000992, A001190, A063834, A196545, A273873, A289501, A300354, A300439, A300442.

Programs

Maple

a:= proc(n) option remember;
      1+add(a(j)*a(n-j), j=1..n/2)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 06 2018

Mathematica

j[n_]:=j[n]=1+Sum[Times@@j/@y,{y,Select[IntegerPartitions[n],Length[#]===2&]}];
Array[j,40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n-j], {j, 1, n/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

PARI

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=1, n\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

Formula

a(n) = 1 + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).