The algebraic equation satisfied by the probability generating fucntion, P=P\
(t), for walks with the probability distribution
{[-2, 1/4], [-1, 1/4], [1, 1/2]}
that start at 0, never go to the negatives, and end anywhere on the non-nega\
tives is
2 2 3
8 + (-8 + 12 t) P + t (-8 + 7 t) P + 2 t (-1 + t) P = 0
and in Maple input notation
8+(-8+12*t)*P+t*(-8+7*t)*P^2+2*t^2*(-1+t)*P^3 = 0
The algebraic equations for those walks that end exactly at, 0, is
2 2 3 3
16 - 16 P + 2 t P + t P = 0
and in Maple input notation
16-16*P+2*t^2*P^2+t^3*P^3 = 0
The algebraic equations for those walks that end exactly at, 1, is
2 2 4 3
-32 t - 16 (t - 2) (t + 2) P - 2 t (8 + t) P + t P = 0
and in Maple input notation
-32*t-16*(t-2)*(t+2)*P-2*t^2*(8+t)*P^2+t^4*P^3 = 0
This took, 3.053, seconds.