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