A268172 Binary-ternary Wedderburn-Etherington numbers.

0, 1, 1, 2, 4, 9, 23, 58, 156, 426, 1194, 3393, 9802, 28601, 84347, 250732, 750908, 2262817, 6857386, 20882889, 63877262, 196162762, 604567254, 1869318719, 5797113028, 18026873112, 56197262814, 175594836698, 549839459963, 1725126992844, 5422602630117, 17074281639963, 53848886560675, 170085320026578

Offset

0,4

Comments

This is the number of non-planar binary-ternary rooted trees (every node has out-degree 0 or 2 or 3) with n leaf nodes, indexed by the number of leaf nodes (NOT the total number of nodes).

It can also be interpreted as the number of bracketings (valid placements of operation symbols) in a monomial of degree n in a nonassociative algebra with an (anti-)commutative binary operation and a completely (skew-)symmetric ternary operation.

Links

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

Murray R. Bremner, Maple code for binary-ternary Wedderburn-Etherington numbers

Murray R. Bremner, Recursion formula for binary-ternary Wedderburn-Etherington numbers

Formula

See Maple code, and the recursion formula under Links.

Example

Here are the 1, 1, 2, 4, 9, 23 bracketings for degrees 1 to 6 (using the monomial interpretation), where the binary and ternary operations are written [-,-] and [-,-,-] respectively, and the hyphen is a placeholder for the argument symbols:
Degree 1: -.
Degree 2: [-,-].
Degree 3: [[-,-],-], [-,-,-].
Degree 4: [[[-,-],-],-], [[-,-],[-,-]], [[-,-,-],-], [[-,-],-,-].
Degree 5:
   [[[[-,-],-],-],-],
   [[[-,-,-],-],-],
   [[[-,-],[-,-]],-],
   [[[-,-],-,-],-],
   [[[-,-],-],[-,-]],
   [[-,-,-],[-,-]],
   [[[-,-],-],-,-],
   [[-,-,-],-,-],
   [[-,-],[-,-],-].
Degree 6:
   [[[[[-,-],-],-],-],-],
   [[[[-,-,-],-],-],-],
   [[[[-,-],[-,-]],-],-],
   [[[[-,-],-,-],-],-],
   [[[[-,-],-],[-,-]],-],
   [[[-,-,-],[-,-]],-],
   [[[[-,-],-],-,-],-],
   [[[-,-,-],-,-],-],
   [[[-,-], [-,-],-],-],
   [[[[-,-],-],-],[-,-]],
   [[[-,-,-],-],[-,-]],
   [[[-,-], [-,-]],[-,-]],
   [[[-,-],-,-],[-,-]],
   [[[-,-],-],[[-,-],-]],
   [[[-,-],-],[-,-,-]],
   [[-,-,-],[-,-,-]],
   [[[[-,-],-],-],-,-],
   [[[-,-,-],-],-,-],
   [[[-,-],[-,-]],-,-],
   [[[-,-],-,-],-,-],
   [[[-,-],-],[-,-],-],
   [[-,-,-],[-,-],-],
   [[-,-],[-,-],[-,-]].

Maple

# for first Maple program see Links
# second Maple program:
b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
      `if`(v=n, 1, add(binomial(a(i)+j-1, j)*
       b(n-i*j, i-1, v-j), j=0..min(n/i, v)))))
    end:
a:= proc(n) option remember; `if`(n<2, n,
      add(b(n, n+1-j, j), j=2..3))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 28 2016

Mathematica

b[n_, i_, v_] := b[n, i, v] = If[n==0, If[v==0, 1, 0], If[i<1 || v<1 || n<v, 0, If[v==n, 1, Sum[Binomial[a[i]+j-1, j]*b[n-i*j, i-1, v-j], {j, 0, Min[n/i, v]}]]]]; a[n_] := a[n] = If[n<2, n, Sum[b[n, n+1-j, j], {j, 2, 3}]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 25 2017, after Alois P. Heinz *)

Crossrefs

Cf. A001190 (Binary Wedderburn-Etherington numbers).

Cf. A000598 (Ternary Wedderburn-Etherington numbers: number of non-planar ternary rooted trees with n nodes): note that this sequence is indexed by the total number of nodes, NOT the number of leaves.

Cf. A000669, A268163.

Column k=3 of A292085.

Sequence in context: A337516 A340920 A337517 * A151404 A027071 A364485

Adjacent sequences: A268169 A268170 A268171 * A268173 A268174 A268175

Keyword

easy,nonn

Author

Murray R. Bremner, Jan 27 2016

Status

approved