### Related

##### facexpand-factorial_expand-makefact-minfactorial-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

##### facexpand-factorial_expand-makefact-minfactorial-rat-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

##### facexpand-factorial_expand-minfactorial-sum

f(k):=(2*n)!/(k!*(2*n...

facexpand: true, fact...

minfactorial(f(k+1)/f...

Calculate

##### facexpand-factorial_expand-makefact-minfactorial-num-rat-subst-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

##### facexpand-factorial_expand-makefact-minfactorial-num-rat-subst-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

##### facexpand-factorial_expand-makefact-minfactorial-num-rat-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

##### facexpand-factorial_expand-makefact-minfactorial-num-rat-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

##### facexpand-factorial_expand-makefact-minfactorial-rat-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

##### facexpand-factorial_expand-makefact-minfactorial-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

##### facexpand-factorial_expand-makefact-minfactorial-num-rat-subst-true

f(n,k):=n!*n!/(n*k!*(...

makefact:true;

facexpand: true, fact...

Calculate

### facexpand

Run Example
```(%i1)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) ```
Run Example
```f(n,k):=1/n*binomial(n,k)*binomial(n,k-1);
1
(%o1)           f(n, k) := - binomial(n, k) binomial(n, k - 1)
n
(%i2) makefact:true;
(%o2)                                true
(%i3) facexpand: true, factorial_expand: true;
(%o3)                                true
(%i4) minfactorial(f(n-1,k)/f(n,k));
n binomial(n - 1, k - 1) binomial(n - 1, k)
(%o4)             -------------------------------------------
(n - 1) binomial(n, k - 1) binomial(n, k)
(%i5) eq: A*minfactorial(f(n-1,k)/f(n,k))+B*minfactorial(f(n-1,k-1)/f(n,k))-1;
n binomial(n - 1, k - 2) binomial(n - 1, k - 1) B
(%o5) -------------------------------------------------
(n - 1) binomial(n, k - 1) binomial(n, k)
n binomial(n - 1, k - 1) binomial(n - 1, k) A
+ --------------------------------------------- - 1
(n - 1) binomial(n, k - 1) binomial(n, k)
(%i6) ```

### Related Help

Help for Facexpand