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factorial_expand

Run Example
(%i1)declare(m,integer);
(%o1)                                done
(%i2) factorial_expand:true;
(%o2)                                true
(%i3) epsilon(n,k):=(binomial(n,k)*gamma(3/2)/gamma(3/2-n+k)-(-1)^n*binomial(n-1,k-1)*n!/k!);
                                            3
                       binomial(n, k) gamma(-)
                                            2
(%o3) epsilon(n, k) := -----------------------
                                3
                          gamma(- - n + k)
                                2
                                                    n
                                               (- 1)  binomial(n - 1, k - 1) n!
                                             - --------------------------------
                                                              k!
(%i4) G(m):=-m*(4*m-3)/2;
                                    (- m) (4 m - 3)
(%o4)                       G(m) := ---------------
                                           2
(%i5) rat(epsilon(2*m,2*m-4)-G(m)*epsilon(2*m-1,2*m-4));
                8         7          6          5          4          3
(%o5)/R/ (1024 m  - 8704 m  + 30976 m  - 59680 m  + 67036 m  - 43756 m
                                                                 2
                                                        + 15309 m  - 2205 m)/96
(%i6) epsilon(4,1)-G(2)*epsilon(3,1);
                                      15
(%o6)                                 --
                                      4
(%i7) epsilon(2,1);
(%o7)                                 - 1
(%i8) epsilon(2,0);
                                        1
(%o8)                                 - -
                                        4
(%i9) 
Run Example
f(n,k):=n!*n!/(n*k!*(n-k)!*(k-1)!*(n-k+1)!);
                                          n! n!
(%o1)           f(n, k) := -----------------------------------
                           n k! (n - k)! (k - 1)! (n - k + 1)!
(%i2) makefact:true;
(%o2)                                true
(%i3) facexpand: true, factorial_expand: true;
(%o3)                                true
(%i4) eq: num(rat(A*minfactorial(f(n-1,k)/f(n,k))+B*minfactorial(f(n-1,k-1)/f(n,k))-1));
               2            2                    2           2
(%o4)/R/     (k  - k) B + (n  + (- 2 k + 1) n + k  - k) A - n  + n
(%i5) eq: rat(expand(eq),n);
                  2                             2            2
(%o5)/R/ (A - 1) n  + ((- 2 k + 1) A + 1) n + (k  - k) B + (k  - k) A
(%i6) subst(a1*n+a2*k+a3,A,eq);
        2
(%o6) (k  - k) B + n ((1 - 2 k) (a1 n + a2 k + a3) + 1)
                          2                            2
                      + (k  - k) (a1 n + a2 k + a3) + n  (a1 n + a2 k + a3 - 1)
(%i7) 
Run Example
f(k):=(2*n)!/(k!*(2*n-k)!)*a^(2*n-k)*b^k;
                                  (2 n)!      2 n - k  k
(%o1)                  f(k) := ------------- a        b
                               k! (2 n - k)!
(%i2) facexpand: true, factorial_expand: true;
(%o2)                                true
(%i3) f(k+1)/f(k);
                                b k! (2 n - k)!
(%o3)                      -------------------------
                           a (k + 1)! (2 n - k - 1)!
(%i4) minfactorial(f(k+1)/f(k));
                                  b (2 n - k)
(%o4)                             -----------
                                   a (k + 1)
(%i5) eq: sum(f(kk),kk,0,n);
                                 n
                                ====     2 n - kk  kk
                                \       a         b
(%o5)                    (2 n)!  >     ---------------
                                /      kk! (2 n - kk)!
                                ====
                                kk = 0
(%i6) solve_rec();
(%o6)                             solve_rec()
(%i7) 

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