A007596 Erroneous version of A226909.
1, 2, 4, 44, 164, 616
Offset: 0
Keywords
Links
- P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. See p. 166, but note that 14 is missing.
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Triangle begins: 1; 1, 1; 1, 2, 1; 2, 5, 5, 2; 3, 11, 16, 11, 3; 6, 26, 50, 50, 26, 6; 11, 60, 143, 188, 143, 60, 11; 23, 142, 404, 656, 656, 404, 142, 23; ...
B[m_] := Module[{u}, u = Table[0, {m}]; u[[1]] = 1; For[n = 1, n <= Length[u] - 1, n++, u[[n + 1]] = (1 + y)*(Sum[u[[i]]*u[[n + 1 - i]], {i, 1, n}] + If[OddQ[n], u[[Quotient[n, 2] + 1]] /. y -> y^2, 0])/2]; u];
CoefficientList[#, y]& /@ B[11] // Flatten (* Jean-François Alcover, Sep 24 2019, from PARI *)
B(n)={my(u=vector(n)); u[1]=1; for(n=1, #u-1, u[n+1]=(1+y)*(sum(i=1, n, u[i]*u[n+1-i]) + if(n%2, subst(u[n\2+1], y, y^2)))/2); u}
{ my(A=B(10)); for(n=1, #A, print(Vec(A[n]))) } \\ Andrew Howroyd, May 21 2018
For n = 4 the 25 association types are as follows, where * is commutative and # is noncommutative; some assumptions have been made regarding the order of the factors for the commutative operation: ( ( X * X ) * X ) * X, ( ( X # X ) * X ) * X, ( ( X * X ) # X ) * X, ( ( X # X ) # X ) * X, ( X # ( X * X ) ) * X, ( X # ( X # X ) ) * X, ( X * X ) * ( X * X ), ( X * X ) * ( X # X ), ( X # X ) * ( X # X ), ( ( X * X ) * X ) # X, ( ( X # X ) * X ) # X, ( ( X * X ) # X ) # X, ( ( X # X ) # X ) # X, ( X # ( X * X ) ) # X, ( X # ( X # X ) ) # X, ( X * X ) # ( X * X ), ( X * X ) # ( X # X ), ( X # X ) # ( X * X ), ( X # X ) # ( X # X ), X # ( ( X * X ) * X ), X # ( ( X # X ) * X ), X # ( ( X * X ) # X ), X # ( ( X # X ) # X ), X # ( X # ( X * X ) ), X # ( X # ( X # X ) ).
BWT := table():
BWT[ 1 ] := 1:
for arity from 2 to 24 do
BWT[ arity ] := 0:
# commutative operation
for i to floor((arity-1)/2) do
BWT[ arity ] := BWT[ arity ] + ( BWT[arity-i] * BWT[i] )
od:
if arity mod 2 = 0 then
BWT[ arity ] := BWT[ arity ] + binomial( BWT[arity/2]+1, 2 )
fi:
# noncommutative operation
for i to arity-1 do
BWT[ arity ] := BWT[ arity ] + ( BWT[arity-i] * BWT[i] )
od
od:
seq(BWT[ n ], n=1..24);
BWT[1] = 1; For[arity = 2, arity <= 24, arity++, BWT[arity] = 0; (* commutative operation *) For[i = 1, i <= Floor[(arity-1)/2], i++, BWT[arity] = BWT[arity] + (BWT[arity-i]*BWT[i])]; If[EvenQ[arity], BWT[arity] = BWT[arity] + Binomial[BWT[ arity/2]+1, 2]]; (* non commutative operation *) For[i = 1, i <= arity-1, i++, BWT[arity] = BWT[arity] + (BWT[arity-i]*BWT[i])]];
Table[BWT[n], {n, 1, 24}] (* Jean-François Alcover, Feb 15 2019, from Maple *)
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