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sumsplitfact

? sumsplitfact;

Calculate

sumsplitfact

? 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) 

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