GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
216 views
Find $\large\color{maroon}{a^n}$ for the following generating function,

$$\color{green}{\begin{align*} \frac{1}{1-2x^2} \end{align*}}$$

$\large\color{maroon}{a^n}$ = closed form of the $nth$ term in the corresponding sequence.
asked in Combinatory by Veteran (50.5k points)  
retagged by | 216 views
<1,2,4,8....> ??

This is what you wanted ??
possibly a closed form

1 Answer

+3 votes
Best answer

$\frac{1}{1-2x^{2}}$ = $\frac{1}{1-\sqrt{2}x}$ * $\frac{1}{1+\sqrt{2}x}$    ................(1)

 

Now, we know that,

$\frac{1}{1-ax}$ = 1+ ax + a2x + a3x3  +..................         ..........(2)

$\frac{1}{1+ax}$ = 1- ax + a2x - a3x3  +..................         ...........(3)

 

Now, from (1) , we observe that the result is the product of 2 functions.

 

If the coefficients of function 1 are a0, a1, a2................

and that of function 2 are b0, b1, b2................

then the result of product of two functions is a function with terms having coefficients rk corresponding to ''k'th term of the sequence. The coefficients can be calculated as follows:

rk = akb0 + ak-1b1 + ak-2b2.........................+ a0bk

 

From (2) and (3), we get,

r0 = 1

r1 = 0

r2 = a2

r3 = 0

........

 

We observe that when 'k' is odd, rk = 0. If 'k' is even, rk = ak

Thus, the terms of the sequence are {1, 0, a2, 0, a4, 0, ........................}

 

Now, substitute $\sqrt{2}$ for 'a' to get the actual sequence.

answered by Veteran (15.2k points)  
selected by
nice answer.


Top Users Jul 2017
  1. Bikram

    5368 Points

  2. manu00x

    3092 Points

  3. Arjun

    1924 Points

  4. joshi_nitish

    1894 Points

  5. Debashish Deka

    1874 Points

  6. Tesla!

    1380 Points

  7. pawan kumarln

    1336 Points

  8. Hemant Parihar

    1314 Points

  9. Shubhanshu

    1136 Points

  10. Arnab Bhadra

    1124 Points


24,137 questions
31,140 answers
70,874 comments
29,458 users