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# Name:Treasa Bency Biju Jose
# Date: 10-06-2020
# Assignment #11
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1. 
Snk(11, 5);
                             246730

Bell1(11);
                             678570

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2.
xn := proc(x, n) local t, i; t := seq(product(x - i, i = 0 .. n - 1)); end proc;

Axn := proc(x, i) local k, t, r, j; t := seq(Snk(i, k)*xn(x, k), k = 1 .. i); r := sum(t[j], j = 1 .. i); end proc;

g := [seq(Axn(x, i), i = 0 .. 20)];
g := [0, x[1], 2 x, 7 x - 3, 21 x - 7, 81 x - 40, 333 x - 301, 

  1380 x - 966, 6555 x - 4726, 32896 x - 32640, 

  165433 x - 233131, 917280 x - 961026, 5468477 x - 5310031, 

  33448689 x - 38509653, 204250413 x - 305175781, 

  1230416470 x - 2368946586, 8924565505 x - 12014156731, 

  66117111757 x - 78263729158, 488241037955 x - 638628998578, 

  3555892367056 x - 5702521065920, 

  25459096260937 x - 50825984897871]

t := [seq(Axn(5, i), i = 0 .. 20)];
t := [0, 5[1], 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 

  9765625, 48828125, 244140625, 1220703125, 6103515625, 

  30517578125, 152587890625, 762939453125, 3814697265625, 

  19073486328125, 95367431640625]


with(gfun);
guessgf(t, x);
              [-((-5*5[1] + 25)*x^2 + 5[1]*x)/(5*x - 1), ogf]

l := [seq(Axn(6, i), i = 0 .. 20)];
  l := [0, 6[1], 36, 216, 1296, 7776, 46656, 279936, 1679616, 

    10077696, 60466176, 362797056, 2176782336, 13060694016, 

    78364164096, 470184984576, 2821109907456, 16926659444736, 

    101559956668416, 609359740010496, 3656158440062976]


guessgf(l, x);
            [-((-6*6[1] + 36)*x^2 + 6[1]*x)/(6*x - 1), ogf]

# Hence the generalised formula for Axn(x,n) for any positive n in terms of y is:
# Axn(x, n) = [-((x^2 - 11*x)*y^2 + xy)/(xy - 1), ogf]

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3. Proving the recurrence c(n,k) = c(n-1,k-1) + (n-1) * c(n-1,k)
# binomial(n,k) := binomial(n-1,k-1) + biomial(n-1,k)
# Cases: n does or does not belong to the set
# S(n,k):= Set partions {1..n} with exactly k cycles

# Case 1: n not part of the set gives => {1..n-1} and k-1 cycles
# Case 2: Since n is not alone, removing it will => {1..n-1} with k cycles

# Hence k has n-1 options existing and so (n-1) * c(n-1,k)

# c(n,k) = c(n-1,k-1) + (n-1) * c(n-1,k)

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4. 
cnk := proc(n, k) option remember; 
if n = 1 then 
  if k = 1 then 
    RETURN(1); 
  else RETURN(0); 
  end if; 
end if; 
cnk(n - 1, k - 1) + (n - 1)*cnk(n - 1, k); 
end proc:

cnk(5, 3);
                               35

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5.
factor(Sum(c(5, 3)*x^k, k = 1 .. 5));
                            5       
                          -----      
                           \         
                            )           k
                           /    35 x 
                          -----      
                          k = 1      

c(5, 3);
                               35

# from above we know that the generalised formula is 


                            n        
                          -----      
                           \         
                            )            k
                           /    c(n,k) x 
                          -----      
                          k = 1      

factor(sum(c(5, 3)*x^k, k = 1 .. 5));
           
                        35*x*(x^4 + x^3 + x^2 + x + 1)

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