A301467 Number of enriched r-trees of size n with no empty subtrees.

1, 2, 4, 8, 20, 48, 136, 360, 1040, 2944, 8704, 25280, 76320, 226720, 692992, 2096640, 6470016, 19799936, 61713152, 190683520, 598033152, 1863995392, 5879859200, 18438913536, 58464724992, 184356152832, 586898946048, 1859875518464, 5941384080384, 18901502482432

Offset

1,2

Comments

An enriched r-tree of size n > 0 with no empty subtrees is either a single node of size n, or a finite nonempty sequence of enriched r-trees with no empty subtrees and with weakly decreasing sizes summing to n - 1.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1910

Formula

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

Example

The a(4) = 8 enriched r-trees with no empty subtrees: 4, (3), (21), ((2)), (111), ((11)), ((1)1), (((1))).
The a(5) = 20 enriched r-trees with no empty subtrees:
  5,
  (4), ((3)), ((21)), (((2))), ((111)), (((11))), (((1)1)), ((((1)))),
  (31), (22), (2(1)), ((2)1), ((1)2), ((11)1), ((1)(1)), (((1))1),
  (211), ((1)11),
  (1111).

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)* a(i)^j, j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, 1+b(n-1$2)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 21 2018

Mathematica

pert[n_]:=pert[n]=If[n===1, 1, 1+Sum[Times@@pert/@y, {y, IntegerPartitions[n-1]}]];
Array[pert, 30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1] a[i]^j, {j, 0, n/i}]]];
a[n_] := a[n] = If[n < 2, n, 1 + b[n-1, n-1]];
Array[a, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

Prog

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

Crossrefs

Cf. A000081, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A300660, A301342-A301345, A301364-A301368, A301422, A301462, A301469, A301470.

Sequence in context: A369496 A225585 A121703 * A358056 A275070 A115219

Adjacent sequences: A301464 A301465 A301466 * A301468 A301469 A301470

Keyword

nonn

Author

Gus Wiseman, Mar 21 2018

Status

approved