#Logarithm of Bar-Natan's rational associator

#1/48*[AB] 
#-1/48*[BA] 

#-1/1440*[A[A[AB]]] 
#+1/1152*[A[B[AB]]]
#+1/1440*[B[B[BA]]] 
#-1/1152*[B[A[BA]]]

#Expansion of the monomials present in BN's ratonal associator
#over the (filtered) Hall basis:

#[A[B[AB]]]
# - AABB + 2*ABAB - 2*BABA + BBAA;
#[0,0,-1,0,2,0,0,0,0,-2,0,1,0,0]; ==> -c22

#(B,(B,(B,A)))
# - ABBB + 3*BABB - 3*BBAB + BBBA;
#[0,0,0,0,0,0,-1,0,0,0,3,0,-3,1]; ==> -c31

#(B,(A,(B,A)))
#AABB - 2*ABAB + 2*BABA - BBAA;
#[0,0,1,0,-2,0,0,0,0,2,0,-1,0,0]; ==> c22

ass4:=-1/1440*c13 + 1/1152*(-c22) +1/1440*(-c31) -1/1152*c22:

ass6:=
+1/60480*c15
#[A[A[A[A[AB]]]]] 
+1/1451520*(c24 + 2*c11c13)
#[A[A[A[B[AB]]]]]
+13/1161216*(c33 + 3*c11c22)
#[A[A[B[B[AB]]]]]
+17/1451520*(c24 + c11c13)
#[A[B[A[A[AB]]]]]
+1/1451520*(c33 + 2*c11c22 - c12c21)
#[A[B[A[B[AB]]]]]
-1/60480*(-c51)
#[B[B[B[B[BA]]]]] 
-1/1451520*(-c42)
#[B[B[B[A[BA]]]]]
-13/1161216*(-c33)
#[B[B[A[A[BA]]]]]
-17/1451520*(- c42 - c11c31)
#[B[A[B[B[BA]]]]]
-1/1451520*(- c33 - c11c22 + c12c21)
#[B[A[B[A[BA]]]]]
:

#(A,(A,(A,(A,(A,B))))) = c15
#(A,(A,(A,(B,(A,B))))) = c24 + 2*c11c13
#(A,(A,(B,(B,(A,B))))) = c33 + 3*c11c22
#(A,(B,(A,(A,(A,B))))) = c24 + c11c13
#(A,(B,(A,(B,(A,B))))) = c33 + 2*c11c22 - c12c21

#(B,(B,(B,(B,(B,A))))) = - c51
#(B,(B,(B,(A,(B,A))))) = - c42
#(B,(B,(A,(A,(B,A))))) = - c33
#(B,(A,(B,(B,(B,A))))) = - c42 - c11c31
#(B,(A,(B,(A,(B,A))))) = - c33 - c11c22 + c12c21

ass4;
ass6;
-ass6*2^6*6^3;