Let denote the path from
the path ends at at time
this means first we visit then move steps away from then move steps from
note the part is fixed
then
Now to our question
that number of paths that reaches a then at time is
the number of path that reach then and finally at time is
so the number of that reach then without touching in between is
Question 2
let be the paths that touch atleast , times include the path and
the total paths from 0 to without touching a or - is
so