A289501 Number of enriched p-trees of weight n.

1, 1, 2, 4, 12, 32, 112, 352, 1296, 4448, 16640, 59968, 231168, 856960, 3334400, 12679424, 49991424, 192890880, 767229952, 2998427648, 12015527936, 47438950400, 191117033472, 760625733632, 3082675150848, 12346305839104, 50223511928832, 202359539335168

Offset

0,3

Comments

An enriched p-tree of weight n is either (case 1) the number n itself, or (case 2) a sequence of two or more enriched p-trees, having a weakly decreasing sequence of weights summing to n.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1588

Formula

O.g.f.: (1/(1-x) + Product_{i>0} 1/(1-a(i)*x^i))/2.

Example

The a(4) = 12 enriched p-trees are:
  4,
  (31), ((21)1), (((11)1)1), ((111)1),
  (22), (2(11)), ((11)2), ((11)(11)),
  (211), ((11)11),
  (1111).

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
    end:
a:= n-> `if`(n=0, 1, 1+b(n, n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 07 2017

Mathematica

a[n_]:=a[n]=1+Sum[Times@@a/@y, {y, Rest[IntegerPartitions[n]]}];
Array[a, 20]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1,
     If[i<1, 0, b[n, i-1] + a[i] b[n-i, Min[n-i, i]]]];
a[n_] := If[n == 0, 1, 1 + b[n, n-1]];
a /@ Range[0, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

Prog

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

Crossrefs

Cf. A052337, A063834, A093637, A196545, A273873, A281145, A300660.

Sequence in context: A216819 A216820 A148194 * A274961 A027695 A148195

Adjacent sequences: A289498 A289499 A289500 * A289502 A289503 A289504

Keyword

nonn

Author

Gus Wiseman, Jul 07 2017

Status

approved