~/tmp[1]> math Mathematica 9.0 for Linux x86 (64-bit) Copyright 1988-2013 Wolfram Research, Inc. In[1]:= SetOptions["stdout",PageWidth->50]; In[2]:= << christmas20.m Out[2]= christmas merry In[3]:= B[2,10] 2 3 4 5 Out[3]= 1 + z + 2 z + 5 z + 14 z + 42 z + 6 7 8 9 10 > 132 z + 429 z + 1430 z + 4862 z + O[z] In[4]:= B[3,9] 2 3 4 5 Out[4]= 1 + z + 3 z + 12 z + 55 z + 273 z + 6 7 8 9 > 1428 z + 7752 z + 43263 z + O[z] In[5]:= B[1,14] 2 3 4 5 6 7 Out[5]= 1 + z + z + z + z + z + z + z + 8 9 10 11 12 13 14 > z + z + z + z + z + z + O[z] In[6]:= B[0,100] 100 Out[6]= 1 + z + O[z] In[7]:= B[3/2,9] 2 3 5 3 z 21 z 4 1287 z Out[7]= 1 + z + ---- + ----- + 5 z + ------- + 2 8 128 7 6 46189 z 8 9 > 21 z + -------- + 99 z + O[z] 1024 In[8]:= B[Pi,9] 2 2 -Pi 3 Pi 3 Out[8]= 1 + z + Pi z + (--- + -----) z + 2 2 2 3 4 (Pi - 6 Pi + 8 Pi ) z > ----------------------- + 3 2 3 4 5 (-6 Pi + 55 Pi - 150 Pi + 125 Pi ) z > --------------------------------------- + 24 6 z Binomial[1 + 6 Pi, 6] > ------------------------ + 1 + 6 Pi 7 z Binomial[1 + 7 Pi, 7] > ------------------------ + 1 + 7 Pi 8 z Binomial[1 + 8 Pi, 8] 9 > ------------------------ + O[z] 1 + 8 Pi In[9]:= B[I,9] 2 3 I 3 Out[9]= 1 + z + I z - (- + -) z + 2 2 7 I 4 35 5 31 I 6 > (2 - ---) z + (-- + 6 I) z - (-- - -) z + 3 12 2 2 1043 279 I 7 194 4393 I 8 9 > (---- - -----) z + (--- + ------) z + O[z] 72 8 3 63 In[10]:= B[3,8]^2 2 3 4 Out[10]= 1 + 2 z + 7 z + 30 z + 143 z + 5 6 7 8 > 728 z + 3876 z + 21318 z + O[z] In[11]:= Sum[Binomial[3k+2,k](2/(3k+2))z^k,{k,0,7}] 2 3 4 Out[11]= 1 + 2 z + 7 z + 30 z + 143 z + 5 6 7 > 728 z + 3876 z + 21318 z In[12]:= g0=B[3/2,16]^(1/2) 2 4 5 z 5 z 3 231 z 7 z Out[12]= 1 + - + ---- + z + ------ + ---- + 2 8 128 2 6 8 9 7293 z 7 1062347 z 143 z > ------- + 15 z + ---------- + ------ + 1024 32768 2 10 12 42010995 z 11 3506302275 z > ------------ + 364 z + -------------- + 262144 4194304 14 13 151973158605 z 15 > 1938 z + ---------------- + 10659 z + 33554432 16 > O[z] In[13]:= FactorInteger[33554432] Out[13]= {{2, 25}} In[14]:= g1=Normal[g0]/.z->-z + O[z]^16 2 4 5 z 5 z 3 231 z 7 z Out[14]= 1 - - + ---- - z + ------ - ---- + 2 8 128 2 6 8 9 7293 z 7 1062347 z 143 z > ------- - 15 z + ---------- - ------ + 1024 32768 2 10 12 42010995 z 11 3506302275 z > ------------ - 364 z + -------------- - 262144 4194304 14 13 151973158605 z 15 > 1938 z + ---------------- - 10659 z + 33554432 16 > O[z] In[15]:= g0 g1 2 4 6 8 Out[15]= 1 + z + 3 z + 12 z + 55 z + 10 12 14 16 > 273 z + 1428 z + 7752 z + O[z] In[16]:= g0-g1 3 5 7 9 Out[16]= z + 2 z + 7 z + 30 z + 143 z + 11 13 15 16 > 728 z + 3876 z + 21318 z + O[z] In[17]:= Simplify[(u-T[2,1,15,0])(u-T[2,1,15,1])(u-1/T[1,2,15,0])] 2 u 3 13 Out[17]= -(--) + (1 + u ) + O[z] z In[18]:= Table[c[p,q],{q,0,9},{p,0,Floor[q/2]}]//MatrixForm Out[18]//MatrixForm= {1} {1} {1, 1} {1, 2} {1, 3, 3} {1, 4, 7} {1, 5, 12, 12} {1, 6, 18, 30} {1, 7, 25, 55, 55} {1, 8, 33, 88, 143} In[19]:= Table[d[p,q],{q,0,9},{p,0,Floor[2q/3]}]//MatrixForm Out[19]//MatrixForm= > {1} {1} {1, 1} {1, 2, 2} {1, 3, 5} {1, 4, 9, 9} {1, 5, 14, 23, 23} {1, 6, 20, 43, 66} {1, 7, 27, 70, 136, 136} {1, 8, 35, 105, 241, 377, 377} In[20]:= T[3,2,15,0] 2 11/3 16/3 7 1/3 z 4 z 65 z 5 z Out[20]= z + -- + ------- + -------- + ---- + 3 9 81 3 26/3 31/3 2737 z 58520 z 12 > ---------- + ----------- + 22 z + 729 6561 41/3 3294980 z 15 > ------------- + O[z] 59049 In[21]:= T[3,2,15,1] 2 1/3 11/3 2/3 1/3 z 4 (-1) z Out[21]= (-1) z + -- - --------------- + 3 9 2/3 16/3 7 65 (-1) z 5 z > ---------------- + ---- - 81 3 1/3 26/3 2/3 31/3 2737 (-1) z 58520 (-1) z > ------------------ + ------------------- + 729 6561 1/3 41/3 12 3294980 (-1) z 15 > 22 z - --------------------- + O[z] 59049 In[22]:= T[3,2,15,2] 2 2/3 11/3 1/3 1/3 z 4 (-1) z Out[22]= -((-1) z ) + -- + --------------- - 3 9 1/3 16/3 7 65 (-1) z 5 z > ---------------- + ---- + 81 3 2/3 26/3 1/3 31/3 2737 (-1) z 58520 (-1) z > ------------------ - ------------------- + 729 6561 2/3 41/3 12 3294980 (-1) z 15 > 22 z + --------------------- + O[z] 59049 In[23]:= 1/T[2,3,15,0] 2 9/2 7 1 z 7 z 5 z Out[23]= ------- - -- - ------ - ---- - Sqrt[z] 2 8 2 19/2 1105 z 12 14 > ---------- - 33 z + O[z] 128 In[24]:= 1/T[2,3,15,1] 2 9/2 7 1 z 7 z 5 z Out[24]= -(-------) - -- + ------ - ---- + Sqrt[z] 2 8 2 19/2 1105 z 12 14 > ---------- - 33 z + O[z] 128 In[25]:= Simplify[(u-T[3,2,15,0])(u-T[3,2,15,1])(u-T[3,2,15,2]) (u-1/T[2,3,15,0])(u-1/T[2,3,15,1])] 3 u 5 27/2 Out[25]= -(--) + (1 + u ) + O[z] z In[26]:= funcs[3,2,17] 5 10 15 Out[26]= 1 + 2 z + 23 z + 377 z + 3 8 13 > (z + 9 z + 136 z ) zz + 6 11 2 16 > (z + 5 z + 66 z ) zz + BigO[z] In[27]:= Table[dd[p,q],{q,0,6},{p,0,Floor[3q/2]}]//MatrixForm Out[27]//MatrixForm= {1} {1, 1} {1, 2, 2, 2} {1, 3, 5, 7, 7} {1, 4, 9, 16, 23, 23, 23} {1, 5, 14, 30, 53, 76, 99, 99} {1, 6, 20, 50, 103, 179, 278, 377, 377, 377} In[28]:= funcs[2,3,17] 5 10 15 Out[28]= 1 + 2 z + 23 z + 377 z + 2 7 12 16 > (z + 7 z + 99 z ) zz + BigO[z] In[29]:= Table[duchon[k],{k,8}] Out[29]= {2, 23, 377, 7229, 151491, 3361598, > 77635093, 1846620581} In[30]:= bizley[3,2,6] 2 3 4 Out[30]= 1 + 2 z + 23 z + 377 z + 7229 z + 5 6 > 151491 z + O[z] In[31]:= hat[1,2,.3] Out[31]= 0.367817 In[32]:= hat[1,2,%] Out[32]= 0.3 In[33]:= hat[1,2,.7] Out[33]= 0.0733719 In[34]:= hat[1,2,%] Out[34]= 0.7