? sumsplitfact;

Calculate

? sumsplitfact;

Calculate

### sumsplitfact

Run Example
```(%i1)p: (n-1)!/((k-1)!*(n-k)!)-k*(n-2)!/((k-1)!*(n-k-1)!)-(n-2)!/((k-2)!*(n-k)!);
(n - 1)!            (n - 2)!             k (n - 2)!
(%o1)    ----------------- - ----------------- - ---------------------
(k - 1)! (n - k)!   (k - 2)! (n - k)!   (k - 1)! (n - k - 1)!
(%i2) q: factcomb(p*(k-2)!*(n-k-1)!/(n-2)!), sumsplitfact:false;
n - 1          1       k
(%o2)                   --------------- - ----- - -----
(k - 1) (n - k)   n - k   k - 1
(%i3) num(q);
n - 1          1       k
(%o3)                   --------------- - ----- - -----
(k - 1) (n - k)   n - k   k - 1
(%i4) rat(q);
(%o4)/R/                              - 1
(%i5) ```
Run Example
```f(n,k):=n!^3/k!^3*(n-k)!^3;
3
n!          3
(%o1)                      f(n, k) := --- (n - k)!
3
k!
(%i2) sumsplitfact:false;
(%o2)                                false
(%i3) p: factcomb(f(n,k)-(n^2+n*k+k^2)/(n-k)^2*f(n-1,k)-f(n-1,k-1));
5        3         3             3
n  (n - k)  (n - 1)!  (n - k - 1)!
(%o3) -------------------------------------------------
3         3  2      4         3      5         3
k  (k - 1)!  n  - 2 k  (k - 1)!  n + k  (k - 1)!
4        3         3             3
2 k n  (n - k)  (n - 1)!  (n - k - 1)!
- -------------------------------------------------
3         3  2      4         3      5         3
k  (k - 1)!  n  - 2 k  (k - 1)!  n + k  (k - 1)!
2  3        3         3             3
k  n  (n - k)  (n - 1)!  (n - k - 1)!
+ -------------------------------------------------
3         3  2      4         3      5         3
k  (k - 1)!  n  - 2 k  (k - 1)!  n + k  (k - 1)!
2         3             3
n  (n - 1)!  (n - k - 1)!
- -------------------------------------------------
3         3  2      4         3      5         3
k  (k - 1)!  n  - 2 k  (k - 1)!  n + k  (k - 1)!
3             3
k n (n - 1)!  (n - k - 1)!
- -------------------------------------------------
3         3  2      4         3      5         3
k  (k - 1)!  n  - 2 k  (k - 1)!  n + k  (k - 1)!
2         3             3
k  (n - 1)!  (n - k - 1)!
- -------------------------------------------------
3         3  2      4         3      5         3
k  (k - 1)!  n  - 2 k  (k - 1)!  n + k  (k - 1)!
3         3             3
(n - k)  (n - 1)!  (n - k - 1)!
- --------------------------------
3
(k - 1)!
(%i4) ```
Run Example
```b2(n,k):=binomial(n,k)^2;
2
(%o1)                     b2(n, k) := binomial (n, k)
(%i2) b3(n,k):=binomial(n,k)^3;
3
(%o2)                     b3(n, k) := binomial (n, k)
(%i3) p2: factcomb(makefact(1+a*binomial(n-1,k)*binomial(n-1,k-1)/b2(n-1,k))), sumsplitfact:false;
a k
(%o3)                              ----- + 1
n - k
(%i4) q2: factcomb(makefact(1+(2-a)*binomial(n-1,k)*binomial(n-1,k-1)/b2(n-1,k-1))), sumsplitfact:false;
a (n - k)   2 (n - k)
(%o4)                     - --------- + --------- + 1
k           k
(%i5) p2:rat(p2);
n + (a - 1) k
(%o5)/R/                         -------------
n - k
(%i6) q2:rat(q2);
(a - 2) n + (- a + 1) k
(%o6)/R/                   - -----------------------
k
(%i7) for i:0 thru 2 do (j: 2-i,print(i,j),print(subst([a=i,b=j],p2)),print(subst([a=i,b=j],q2)));
0 2
1
k - 2 n
- -------
k
1 1
n
-----
n - k
n
-
k
2 0
n + k
-----
n - k
1
(%o7)                                done
(%i8) p3: factcomb(makefact(1+a*b2(n-1,k)*binomial(n-1,k-1)/b3(n-1,k)+b*binomial(n-1,k)*b2(n-1,k-1)/b3(n-1,k))), sumsplitfact:false;
2
a k      b k
(%o8)                        ----- + -------- + 1
n - k          2
(n - k)
(%i9) q3: factcomb(makefact(1+(3-a)*b2(n-1,k)*binomial(n-1,k-1)/b3(n-1,k)+(3-b)*binomial(n-1,k)*b2(n-1,k-1)/b3(n-1,k))), sumsplitfact:false;
2          2
a k     3 k      b k        3 k
(%o9)              - ----- + ----- - -------- + -------- + 1
n - k   n - k          2          2
(n - k)    (n - k)
(%i10) p3:rat(p3);
2                              2
n  + (a - 2) k n + (b - a + 1) k
(%o10)/R/              ---------------------------------
2            2
n  - 2 k n + k
(%i11) q3:rat(q3);
2                                  2
n  + (- a + 1) k n + (- b + a + 1) k
(%o11)/R/            -------------------------------------
2            2
n  - 2 k n + k
(%i12) ```

### Related Help

Help for Sumsplitfact