: 5. An open problem. : 0. Abstract : 3. The answer to the previous...

4. Related facts.

There is also a very interesting fact of the binomial Theorem that is closely related our results. This fact was found by us, and presented in R. Miyadera and Y. Kotera [1].

Example 7. The expansion of $ (a + b)^{40} $ . If we expand $ (a + b)^{40} $, then we will find a beautiful curve.

  $\displaystyle a^{40}$    
  $\displaystyle 40 a^{39} b^{1}$    
  $\displaystyle 78 0 a^{38} b^{2}$    
  $\displaystyle 9880 a^{37} b^{3}$    
  $\displaystyle 91390 a^{36} b^{4}$    
  $\displaystyle 658008 a^{35} b^{5}$    
  $\displaystyle 3838380 a^{34} b^6$    
  $\displaystyle 18643560 a^{33} b^7$    
  $\displaystyle 76904685 a^{32} b^8$    
  $\displaystyle 273438880 a^{31} b^9$    
  $\displaystyle 847660528 a^{30} b^{10}$    
  $\displaystyle 2311801440 a^{29} b^{11}$    
  $\displaystyle 5586853480 a^{28} b^{12}$    
  $\displaystyle 12033222880 a^{27} b^{13}$    
  $\displaystyle 23206929840 a^{26} b^{14}$    
  $\displaystyle 40225345056 a^{25} b^{15}$    
  $\displaystyle 62852101650 a^{24} b^{16}$    
  $\displaystyle 88732378800 a^{23} b^{17}$    
  $\displaystyle 113380261800 a^{22} b^{18}$    
  $\displaystyle 131282408400 a^{21} b^{19}$    
  $\displaystyle 137846528820 a^{20} b^{20}$    
  $\displaystyle 131282408400 a^{19} b^{21}$    
  $\displaystyle 113380261800 a^{18} b^{22}$    
  $\displaystyle 88732378800 a^{17} b^{23}$    
  $\displaystyle 62852101650 a^{16} b^{24}$    
  $\displaystyle 40225345056 a^{15} b^{25}$    
  $\displaystyle 23206929840 a^{14} b^{26}$    
  $\displaystyle 12033222880 a^{13} b^{27}$    
  $\displaystyle 5586853480 a^{12} b^{28}$    
  $\displaystyle 2311801440 a^{11} b^{29}$    
  $\displaystyle 847660528 a^{10} b^{30}$    
  $\displaystyle 273438880 a^{9} b^{31}$    
  $\displaystyle 76904685 a^{8} b^{32}$    
  $\displaystyle 18643560 a^{7} b^{33}$    
  $\displaystyle 3838380 a^{6} b^{34}$    
  $\displaystyle 658008 a^{5} b^{35}$    
  $\displaystyle 91390 a^{4} b^{36}$    
  $\displaystyle 9880 a^{3} b^{37}$    
  $\displaystyle 780 a^{2} b^{38}$    
  $\displaystyle 40 a^{1} b^{39}$    
  $\displaystyle b^{40}$    

If we give colours to each binomial coefficient by the same way we used in Example 3, then we get the following picture.

\includegraphics[height=6cm,clip]{graph9-17.eps}

Theorem 2. The shapes of the expansion of $ (a + b)^n $ divided by n will converge to the graph of the funcion $ f(x) = -( x \log_{10} x + (1 - x) \log_{10} (1 - x) ) $ , as $ n \to \infty $ .
For proof see Miyadera and Kotera[1].

: 5. An open problem. : 0. Abstract : 3. The answer to the