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# Name:Treasa Bency Biju Jose
# Date: 11-3-2020
# Assignment #17
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1. 
nops(KiTours(3, 50));
                               2

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2.
Findrec(SeqGW({[0, 1], [1, 0], [1, 1], [1, 2], [2, 1]}, 3000), n, N, 3000);
                                             2              3     
  2 (2 n + 3)   (11 n + 23) N   (13 n + 41) N    5 (n + 3) N     4
- ----------- - ------------- - -------------- - ------------ + N 
     n + 5          n + 5           n + 5           n + 5         


ope := -2*(2*n + 3)/(n + 5) - (11*n + 23)*N/(n + 5) - (13*n + 41)*N^2/(n + 5) - 5*(n + 3)*N^3/(n + 5) + N^4;
                                                         2
              2 (2 n + 3)   (11 n + 23) N   (13 n + 41) N 
     ope := - ----------- - ------------- - --------------
                 n + 5          n + 5           n + 5     

                     3     
          5 (n + 3) N     4
        - ------------ + N 
             n + 5         


SeqFromRec(ope, n, N, [2, 8, 37, 187], 3000);
          [Length of output exceeds limit of 1000000]

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3.
ope := Findrec(SeqGW({[0, 0, 1], [0, 1, 0], [1, 0, 0], [1, 1, 1]}, 100), n, N, 3000);
                                             /   2           \  
         (n - 1) (n - 3) (3 n + 8)   (n + 3) \9 n  - 3 n - 16/ N
ope := - ------------------------- + ---------------------------
         (3 n + 5) (n + 5) (n + 4)    (3 n + 5) (n + 5) (n + 4) 

                 /    2            \  2     
     2 (3 n + 7) \15 n  + 65 n + 62/ N     3
   - ---------------------------------- + N 
         (3 n + 5) (n + 5) (n + 4)          


SeqFromRec(ope, n, N, [2, 10, 88], 3000);
          [Length of output exceeds limit of 1000000]

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4.
ope := Z(binomial(n, k)^2*binomial(n + k, k)^2, k, n, N);
              3   /    2            \                      3  2
ope := (n + 1)  - \17 n  + 51 n + 39/ N (2 n + 3) + (n + 2)  N 

A := n1 -> local k; add(binomial(n1, k)^2*binomial(n1 + k, k)^2, k = 0 .. n1);
A := proc (n1) local k; options operator, arrow; add(combinat:-b\

  inomial(n1, k)^2*combinat:-binomial(n1+k, k)^2, k = 0 .. n1) 

   end proc


[seq(A(i), i = 1 .. 2)];
                            [5, 73]

SeqFromRec(ope, n, N, [5, 73], 3000);
          [Length of output exceeds limit of 1000000]

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5.
a := GF3tx(t, x);
                            1                        / 2  2 /  
a := ----------------------------------------------- \x  t  \4 
     / 6  4    5  3    4  3    3  2    \ / 6  4    \           
     \t  x  + t  x  + t  x  + t  x  - 1/ \t  x  - 1/           

   15  9      14  9      14  8       13  8      13  7      12  8
  t   x  + 4 t   x  + 8 t   x  + 12 t   x  + 6 t   x  + 8 t   x 

        12  7      11  7      11  6      10  7       10  6
   + 8 t   x  + 8 t   x  - 2 t   x  + 6 t   x  - 16 t   x 

        9  6       9  5       8  5       8  4       7  4
   - 2 t  x  - 26 t  x  - 26 t  x  - 19 t  x  - 38 t  x 

        7  3       6  4       6  3       5  3    5  2      4  3
   - 7 t  x  - 19 t  x  - 13 t  x  - 13 t  x  - t  x  - 7 t  x 

         4  2    3  2       3         2        2           \\
   + 10 t  x  - t  x  + 16 t  x + 16 t  x + 9 t  + 18 t + 9//


coeff(expand(coeff(taylor(a, x = 0, 150), x, 50)), t, 75);
                          40375223374

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6.
SimuAvDegree(Bn(9)[1], 257, 1000);
                   [4507.587549, 4503.079963]

SimuAvDegree(Bn(9)[1], 257, 1000);
                   [4505.260700, 4500.755440]

SimuAvDegree(Bn(9)[1], 257, 1000);
                   [4505.237354, 4500.732118]

SimuAvDegree(Bn(9)[1], 257, 1000);
                   [4506.638132, 4502.131495]

SimuAvDegree(Bn(9)[1], 257, 1000);
                   [4507.688716, 4503.181029]

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7. 
t := DiagSeq2(1/(11*x*y - 4*x - 5*y + 1), x, y, 200);
t[200];
9708750898075579653384196838671127918874836758262091998689400656\

  08913203043712848903832560751107995551486168394719262425252372\

  36019066093979949706097645202714608598808756891132677394406054\

  78685259685294475900853145417907948009761409228285580391754820\

  91842209534826765111492542089431368664128512648594223941503989\

  788404463487985129890154372129288001


f := 1/(11*x*y - 4*x - 5*y + 1);
coeff(taylor(coeff(taylor(f, x = 0, 201), x, 200), y = 0, 201), y, 200);
                                 1           
                  f := ----------------------
                       11 x y - 4 x - 5 y + 1

9708750898075579653384196838671127918874836758262091998689400656\

  08913203043712848903832560751107995551486168394719262425252372\

  36019066093979949706097645202714608598808756891132677394406054\

  78685259685294475900853145417907948009761409228285580391754820\

  91842209534826765111492542089431368664128512648594223941503989\

  788404463487985129890154372129288001

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