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minfactorial

f(n):=(b+n)!/(a+n)!;

f(n+1)/f(n);

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

Calculate

minfactorial

? minfactorial;

Calculate

minfactorial

n!/(n+2)!;

minfactorial (%);

Calculate

minfactorial

f(n):=(b+n)!/(a+n)!;

f(n+1)/f(n);

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

Calculate

minfactorial

? minfactorial;

Calculate

minfactorial

n!/(n+2)!;

minfactorial (%);

Calculate

minfactorial

Run Example
(%i1)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
P:14760*k^4+28010*k^3+19505*k^2+5920*k+672;
                        4          3          2
(%o1)            14760 k  + 28010 k  + 19505 k  + 5920 k + 672
(%i2) c:(-1)^(k-1)*P/(k^5*(2*k+1)^5);
                  4          3          2                      k - 1
          (14760 k  + 28010 k  + 19505 k  + 5920 k + 672) (- 1)
(%o2)     ----------------------------------------------------------
                                 5          5
                                k  (2 k + 1)
(%i3) minfactorial(N!/(N+1)!);
                                       1
(%o3)                                -----
                                     N + 1
(%i4) minfactorial(binomial(2*k+2,k+1)/binomial(2*k,k));
                           binomial(2 k + 2, k + 1)
(%o4)                      ------------------------
                               binomial(2 k, k)
(%i5) R:ratsimp(subst(k+1,k,c)/c*((k+1)/(4*k+2))^5);
                9          8           7           6          5
(%o5) - (14760 k  + 87050 k  + 192095 k  + 188000 k  + 68867 k )
            9              8              7               6               5
/(15114240 k  + 142039040 k  + 575160320 k  + 1311316480 k  + 1846153728 k
               4              3              2
 + 1656443520 k  + 943453440 k  + 328341600 k  + 63452160 k + 5225472)
(%i6) factor(num(R));
              5         4          3           2
(%o6)      - k  (14760 k  + 87050 k  + 192095 k  + 188000 k + 68867)
(%i7) factor(denom(R));
                     5         4          3          2
(%o7)    32 (2 k + 3)  (14760 k  + 28010 k  + 19505 k  + 5920 k + 672)
(%i8) 
Run Example
P:14760*k^4+28010*k^3+19505*k^2+5920*k+672;
                        4          3          2
(%o1)            14760 k  + 28010 k  + 19505 k  + 5920 k + 672
(%i2) c:(-1)^(k-1)*P/(k^5*(2*k+1)^5);
                  4          3          2                      k - 1
          (14760 k  + 28010 k  + 19505 k  + 5920 k + 672) (- 1)
(%o2)     ----------------------------------------------------------
                                 5          5
                                k  (2 k + 1)
(%i3) minfactorial(binomial(2*k+2,k+1)/binomial(2*k,k));
                           binomial(2 k + 2, k + 1)
(%o3)                      ------------------------
                               binomial(2 k, k)
(%i4) R:ratsimp(subst(k+1,k,c)/c*((k+1)/(4*k+2))^5);
                9          8           7           6          5
(%o4) - (14760 k  + 87050 k  + 192095 k  + 188000 k  + 68867 k )
            9              8              7               6               5
/(15114240 k  + 142039040 k  + 575160320 k  + 1311316480 k  + 1846153728 k
               4              3              2
 + 1656443520 k  + 943453440 k  + 328341600 k  + 63452160 k + 5225472)
(%i5) factor(num(R));
              5         4          3           2
(%o5)      - k  (14760 k  + 87050 k  + 192095 k  + 188000 k + 68867)
(%i6) factor(denom(R));
                     5         4          3          2
(%o6)    32 (2 k + 3)  (14760 k  + 28010 k  + 19505 k  + 5920 k + 672)
(%i7) 

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