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 A268172 Binary-ternary Wedderburn-Etherington numbers. 4
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 11 07:51 EDT 2024. Contains 375059 sequences. (Running on oeis4.)