#Oops! Moves of type  beta  should be split into such a beta (but with
#probability (2*k)/(m+2*k) * 1/(m+2*k-1)  )  in which the last two match "cold"
#and a  delta  changing from state  [m,k]  to  [m,k-1]  in TWO moves, with
#probability (2*k)/(m+2*k) * m/(m+2*k-1)  )


#Maple program.
NN:=18: # upper bound on number of pairs in game
MM:=2*NN: #max number of turns
A:=array[0..NN,0..NN,0..MM]:
for m from 0 to NN do for k from 0 to NN do for s from 0 to MM do 
	A[m,k,s]:=0:od:od:od:
for m from 0 to NN do A[m,0,m]:=1:od: #Can only use moves of type  alpha
A[0,1,1]:=1: 
for m from 1 to NN do #Can only use alphas and one beta or delta if k=1.
  for s from 2 to m+2 do
    A[m,1,s]:=(m/(m+2*1))*A[m-1,1,s-1]
            +(2/(m+2)/(m+1))*A[m,0,s-1]
            +(2*m/(m+2)/(m+1))*A[m,0,s-2]:
  od:
od:
for k from 2 to NN do 
  for m from 0 to NN-k do
    for s from m+k to m+2*k do
        A[m,k,s]:=(m/(m+2*k))*A[m-1,k,s-1]
                +(2*k/(m+2*k)/(m+2*k-1))*A[m,k-1,s-1]
                +(2*k*m/(m+2*k)/(m+2*k-1))*A[m,k-1,s-2]
                +(4*k*(k-1)/(m+2*k)/(m+2*k-1))*A[m+2,k-2,s-1]:
    od:
  od:
od: