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A268172 Binary-ternary Wedderburn-Etherington numbers.

%I A268172 #35 Sep 11 2017 16:38:51
%S A268172 0,1,1,2,4,9,23,58,156,426,1194,3393,9802,28601,84347,250732,750908,
%T A268172 2262817,6857386,20882889,63877262,196162762,604567254,1869318719,
%U A268172 5797113028,18026873112,56197262814,175594836698,549839459963,1725126992844,5422602630117,17074281639963,53848886560675,170085320026578
%N A268172 Binary-ternary Wedderburn-Etherington numbers.
%C A268172 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).
%C A268172 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.
%H A268172 Alois P. Heinz, <a href="/A268172/b268172.txt">Table of n, a(n) for n = 0..1000</a>
%H A268172 Murray R. Bremner, <a href="/A268172/a268172.txt">Maple code for binary-ternary Wedderburn-Etherington numbers</a>
%H A268172 Murray R. Bremner, <a href="/A268172/a268172.pdf">Recursion formula for binary-ternary Wedderburn-Etherington numbers</a>
%F A268172 See Maple code, and the recursion formula under Links.
%e A268172 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:
%e A268172 Degree 1: -.
%e A268172 Degree 2: [-,-].
%e A268172 Degree 3: [[-,-],-], [-,-,-].
%e A268172 Degree 4: [[[-,-],-],-], [[-,-],[-,-]], [[-,-,-],-], [[-,-],-,-].
%e A268172 Degree 5:
%e A268172    [[[[-,-],-],-],-],
%e A268172    [[[-,-,-],-],-],
%e A268172    [[[-,-],[-,-]],-],
%e A268172    [[[-,-],-,-],-],
%e A268172    [[[-,-],-],[-,-]],
%e A268172    [[-,-,-],[-,-]],
%e A268172    [[[-,-],-],-,-],
%e A268172    [[-,-,-],-,-],
%e A268172    [[-,-],[-,-],-].
%e A268172 Degree 6:
%e A268172    [[[[[-,-],-],-],-],-],
%e A268172    [[[[-,-,-],-],-],-],
%e A268172    [[[[-,-],[-,-]],-],-],
%e A268172    [[[[-,-],-,-],-],-],
%e A268172    [[[[-,-],-],[-,-]],-],
%e A268172    [[[-,-,-],[-,-]],-],
%e A268172    [[[[-,-],-],-,-],-],
%e A268172    [[[-,-,-],-,-],-],
%e A268172    [[[-,-], [-,-],-],-],
%e A268172    [[[[-,-],-],-],[-,-]],
%e A268172    [[[-,-,-],-],[-,-]],
%e A268172    [[[-,-], [-,-]],[-,-]],
%e A268172    [[[-,-],-,-],[-,-]],
%e A268172    [[[-,-],-],[[-,-],-]],
%e A268172    [[[-,-],-],[-,-,-]],
%e A268172    [[-,-,-],[-,-,-]],
%e A268172    [[[[-,-],-],-],-,-],
%e A268172    [[[-,-,-],-],-,-],
%e A268172    [[[-,-],[-,-]],-,-],
%e A268172    [[[-,-],-,-],-,-],
%e A268172    [[[-,-],-],[-,-],-],
%e A268172    [[-,-,-],[-,-],-],
%e A268172    [[-,-],[-,-],[-,-]].
%p A268172 # for first Maple program see Links
%p A268172 # second Maple program:
%p A268172 b:= proc(n, i, v) option remember; `if`(n=0,
%p A268172       `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
%p A268172       `if`(v=n, 1, add(binomial(a(i)+j-1, j)*
%p A268172        b(n-i*j, i-1, v-j), j=0..min(n/i, v)))))
%p A268172     end:
%p A268172 a:= proc(n) option remember; `if`(n<2, n,
%p A268172       add(b(n, n+1-j, j), j=2..3))
%p A268172     end:
%p A268172 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jan 28 2016
%t A268172 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_ *)
%Y A268172 Cf. A001190 (Binary Wedderburn-Etherington numbers).
%Y A268172 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.
%Y A268172 Cf. A000669, A268163.
%Y A268172 Column k=3 of A292085.
%K A268172 easy,nonn
%O A268172 0,4
%A A268172 _Murray R. Bremner_, Jan 27 2016