#f(m):=B[m,k,t], Q[k,t]=B[m,k,t] as function of  m.
#Q:=array[0..10,0..10]:
#for i from 0 to 10 do for j from 0 to 10 do Q[i,j]:=0:od:od:
#Q[0,0]:=1:
#Q[1,0]:=2/(m+2):   Q[1,1]:=1-Q[1,0]:
#Q[2,0]:=4/(m+3)/(m+4): Q[2,1]:=4*(5*m+6)/3/(m+3)/(m+4):  Q[2,2]:=1-Q[2,0]-Q[2,1]:
#Q[3,0]:=8/(m+4)/(m+5)/(m+6):
#k:=3:t:=1:
#Assume  k >=2, t>=1. Then
XX:=rsolve({f(m)=(m/(m+2*k))*f(m-1)
                +(2*k/(m+2*k)/(m+2*k-1))*Q[k-1,t]
                +(2*k*m/(m+2*k)/(m+2*k-1))*Q[k-1,t-1]
                +(4*k*(k-1)/(m+2*k)/(m+2*k-1))*subs(m=m+2,Q[k-2,t-1]),
f(0)=subs(m=0,Q[k-1,t])/(2*k-1) + subs(m=2,Q[k-2,t-1])*2*(k-1)/(2*k-1) }
,f(m) );
Q[k,t]:=factor(");