On the average of the number of the Maximal (and Minimal) Number of Balls upon throwing, 1, balls into , 2, bins T times



                              By Shalosh B. Ekhad



 Suppose you throw, uniformly at random,, 1, balls into , 2, boxes  , T  times

     Let , A(T), be the average of the Maximum number of balls minus , T/2



Then A(T) satisfies the following linear recurrence equation with polynomial\
     coefficients



                                  A(T - 1)
                           A(T) = -------- + A(T - 2)
                                   T - 1



                       Subject to the initial conditions



                             A(1) = 1/2, A(2) = 1/2



                             and in Maple notation



A(T) = 1/(T-1)*A(T-1)+A(T-2)


                Using this recurrence we can compute many terms.



                                                                    1/2
         This enables us to estimate that A(T) is asympotically, C T



                        where C is approximately, 0.3989



 Suppose you throw, uniformly at random,, 1, balls into , 2, boxes  , T  times

     Let , A(T), be the average of the Minimum number of balls minus , T/2



Then A(T) satisfies the following linear recurrence equation with polynomial\
     coefficients



                                  A(T - 1)
                           A(T) = -------- + A(T - 2)
                                   T - 1





                       Subject to the initial conditions



                            A(1) = -1/2, A(2) = -1/2

                             and in Maple notation



A(T) = 1/(T-1)*A(T-1)+A(T-2)


                Using this recurrence we can compute many terms.



                                                                    1/2
         This enables us to estimate that A(T) is asympotically, C T



                       where C is approximately, -0.3989



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