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If the ordinary generating function of a sequence $\left \{a_n\right \}_{n=0}^\infty$  is $\large \frac{1+z}{(1-z)^3}$, then $a_3-a_0$ is equal to ___________ .

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$\sum_{n=0}^{\infty }a_{n}z^n=(1+z) \sum_{n=0}^{\infty }\ ^{3-1+n}C_{n}z^{n}$

$\sum_{n=0}^{\infty }a_{n}z^n=(1+z) \sum_{n=0}^{\infty }\ ^{2+n}C_{n}z^{n}$

$\sum_{n=0}^{\infty }a_{n}z^n=\sum_{n=0}^{\infty }\ ^{2+n}C_{n}z^{n}+z\sum_{n=0}^{\infty }\ ^{2+n}C_{n}z^{n}$

$for\ a_{3}:put\ n=3+ put\ n=2$

$a_{3}=\ ^{5}C_{3}+\ ^{4}C_{2}=16$

$for\ a_{0}:put\ n=0+ can't\ provide\ coefficient\ for\ a_{0}\ because\min\ z\ is\ always\ present$

$a_{0}=\ ^{2}C_{0}=1$

$a_{3}-a_{0}=16-1=15$

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can you please elaborate this sir ?

$\frac{1+z}{(1-z)^3} = (1+z)(1-z)^{-3}$

$(1-z)^{-3} = 1 + \binom{3}{1}z + \binom{4}{2}z^2 + \binom{5}{3}z^3 + \dots \infty$

$\color{navy}{(1+z)(1-z)^{-3} = (1+z)*(1 + \binom{3}{1}z + \binom{4}{2}z^2 + \binom{5}{3}z^3 + \dots \infty)}$

$a_0$ is the first term in the expansion of above series and $a_3$ is the fourth term (or) coefficient of $z^3$

$a_0$ = coefficient of $z^0 = 1$
$a_3$ = coefficient of $z^3 = \binom{5}{3} + \binom{4}{2} = 10 + 6$

$\color{maroon}{\Rightarrow a_3 - a_0 = 16 - 1 = 15}$
by Boss
selected
+1
(1−z)−3=1+(3c1)z+(4c2)z2+(5c3)z3+…∞

from where this formula come  ?
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For the coefficient of z3=($\binom{5}{3}$)+($\binom{4}{2}$) = 10+6 .. why are you taking both the terms? why co-eff of z2 as well as z3 are you evaluating?

Won't only z3 co-eff do?

I know I'm wrong somewhere. But couldn't find where?

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plz suggest some good resource to read this topic
+2

This is a good playlist for Discrete u can follow this topic from here or u can follow keneth rosen https://www.youtube.com/playlist?list=PL0862D1A947252D20

+13

Generating functions

Ans : 15

by
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can you explain the step after 'solving'
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Open the brackets and multiply (1+z) to the terms inside. Simple calculation.
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this is the correct formula.

yes 15 is correct ans

by Loyal

Given any sequence say (a0,a1,a2,a3,....) we represent in generating functions as

a0 + a1.z + a2.z2 + a3.z3 + a4.z4 + ....

where z is called as indicator variable.

Now given in question that

a0 + a1.z + a2.z2 + a3.z3 + a4.z4 + ....

= (1 + z) / (1 - z)3

= (1 + z).( 1 / (1-z)3 )

= (1 + z).(1 / 1-z)

= (1 + z).(1 + z + z2 + z3 +z4 + ....)3

= 1.(1 + z + z2 + z3 +z4 + ....)3 + z.(1 + z + z2 + z3 +z4 + ....)3 ==> eqn 1

From eqn 1, I can say that a0 = coefficent of z0 and a3 = coefficient of z3

1.(1 + z + z2 + z3 +z4 + ....)3

= 1.(1 + z + z2 + z3 +z4 + ....).(1 + z + z2 + z3 +z4 + ....).(1 + z + z2 + z3 +z4 + ....)

coefficient of z0 = 1 (Because z0 is possible only if 1 is taken in all 3 bracket terms)

coefficient of z3 = number of ways of choosing (1,z,z2) + (1,1,z3) + (z,z,z)

= (3 * 2) + (3C2 ) + 1

= 6 + 3 + 1

= 10

z.(1 + z + z2 + z3 +z4 + ....)3

= z.(1 + z + z2 + z3 +z4 + ....).(1 + z + z2 + z3 +z4 + ....).(1 + z + z2 + z3 +z4 + ....)

coefficient of z0 = 0 (as we have a z outside and z0 is not possible.

coefficient of z3 = coefficient of z2 from bracket terms

= number of ways choosing (1,1,z2) + (1,z,z)

= 3C2 + 3C2

= 6

a0 = Total coefficient of z0 =  1 + 0 = 1

a3 = Total coefficent of z3 = 10 + 6

= 16

a3 - a0 = 16 - 1 = 15.

by Loyal
edited
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$\frac {(1+z)}{(1-z)^3}$=$(1+z) \ \sum_{r=o}^{\infty} \binom{3-1+r}{r} z^r$   ( $\because$as we know $\frac{1}{(1-z)^n}= \sum_{r=o}^{\infty} \binom{n-1+r}{r} z^r$)

=$(1+z)\binom{2+r}{r} z^r$

=$\binom{2+r}{r} z^r +\binom{2+r}{r} z^{r+1}$

$a_3\ means \ coefficient \ of \ z^3, a_0\ means \ coefficient \ of \ z^0$

for $a_3$ in first term r =3,in second term r=2

$a_3$=$\binom{5}{3}+\binom{4}{2}$=16

for $a_0$ only first term can give $x^0$ coeff. $a_0=\binom{2+0}{0}$=1

$a_3-a_0$=16-1=15.
by Boss
0
nice explanation, thanks.
0
Perfect!!

Given generating function $G(z) = \dfrac{1 + z}{(1-z)^{3}}$

We know that if given sequence is

$(1,1,1,1,1,1,\dots) \Leftrightarrow \dfrac{1}{1-z}$

$\left[\text{Infinite series summation}\: S = \dfrac{a}{1-r}; r< 1\: \text{(or)}\: S = \dfrac{a}{r-1};r > 1\: \text{Where a = first term and r = common ratio} \right]$

$(1 + z + z^{2} + z^{3} + z^{4} + \dots) \Leftrightarrow \dfrac{1}{1-z}$

Differentiate both side with respect to $z$

$\dfrac{\mathrm{d} }{\mathrm{d} z} (1 + z + z^{2} + z^{3} + z^{4} + \dots) \Leftrightarrow \dfrac{\mathrm{d} }{\mathrm{d}z } \left(\dfrac{1}{1-z}\right)$

$(0+1+2z+3z^{2}+4z^{3}+\dots) \Leftrightarrow \dfrac{1}{(1-z)^{2}}$

$(1+2z+3z^{2}+4z^{3}+\dots) \Leftrightarrow \dfrac{1}{(1-z)^{2}}$

Multiply both side by $z$

$(z+2z^{2}+3z^{3}+4z^{4}+\dots) \Leftrightarrow \dfrac{z}{(1-z)^{2}}$

Again differentiate both side with respect to $z$

$(1+4z+9z^{2}+16z^{3}+\dots) \Leftrightarrow \dfrac{(1-z)^{2}(1) - z(2(1-z))(-1)}{((1-z)^{2})^{2}}$

$(1+4z+9z^{2}+16z^{3}+\dots) \Leftrightarrow \dfrac{1+z^{2}-2z +2z(1-z)}{(1-z)^{4}}$

$(1+4z+9z^{2}+16z^{3}+\dots) \Leftrightarrow\dfrac{1+z^{2}-2z +2z-2z^{2}}{(1-z)^{4}}$

$(1+4z+9z^{2}+16z^{3}+\dots) \Leftrightarrow \dfrac{1+z^{2}-2z^{2}}{(1-z)^{4}}$

$(1+4z+9z^{2}+16z^{3}+\dots) \Leftrightarrow \dfrac{1^{2}-z^{2}}{(1-z)^{4}}$

$(1+4z+9z^{2}+16z^{3}+\dots) \Leftrightarrow\dfrac{(1-z)(1+z)}{(1-z)^{4}}$

$(1+4z+9z^{2}+16z^{3}+\dots) \Leftrightarrow \dfrac{(1+z)}{(1-z)^{3}} = G(z)$

$G(z) = 1+4z+9z^{2}+16z^{3}+\dots\:\:\:\rightarrow(1)$

Ordinary Generating Function

$G(z) = a_{0} + a_{1}z +a_{2}z^{2} +a_{3}z^{3}+a_{4}z^{4}+\dots\:\:\:\rightarrow(2)$

Compare equation $(1)$ and equation $(2)$ and we get

$a_{0} =1$

$a_{3}=16$

Now we can easily find $a_{3} -a_{0}=16-1=15$

So, the correct answer is $15.$

See this pdf for generating function it might be helpful.

see here

______________________________________________________________________

$\textbf{Second Method:}$

• $1+x+x^{2}+x^{3}+\dots + x^{n} = \dfrac{1-x^{n+1}}{1-x}$
• $\dfrac{1}{(1-x)^{n}} = \displaystyle{}\sum^{\infty}_{0} \binom{n+k-1}{k}\:x^{k}$

Given generating function $G(z) = \dfrac{1 + z}{(1-z)^{3}}$

$\implies G(z) = (1+z)\:\displaystyle{}\sum^{\infty}_{0} \binom{3+k-1}{k}\:z^{k}$

$\implies G(z) = (1+z)\:\displaystyle{}\sum^{\infty}_{0} \binom{2+k}{k}\:z^{k}$

$\implies G(z) = (1+z)\left[1+3z+6z^{2} + 10z^{3} + \dots \right]$

$\implies G(z) = 1+3z+6z^{2} + 10z^{3} + \dots + z+3z^{2}+6z^{3} + 10z^{4} + \dots$

$\implies G(z) = 1+4z+9z^{2} + 16z^{3} + \dots$

Ordinary Generating Function

$G(z) = a_{0} + a_{1}z +a_{2}z^{2} +a_{3}z^{3}+a_{4}z^{4}+\dots\:\:\:\rightarrow(1)$

$a_{0} =1 ,a_{3}=16$

$\therefore a_{3} -a_{0}=16-1=15$

So, the correct answer is $15.$

by Veteran
edited

Ans:15

by Active

If we have given the ordinary generating function then we can use the following approach to find the sequence $\{a_n\}^{\infty}_{n=0}.$ It is a systematic approach and applicable to any type of generating function.

We can write the Taylor series about origin for the given generating function as :

$f(x) = a_{0} x^{0} + a_{1} x^{1} +a_{2} x^{2}+a_3 x^3+........= \sum_{n}^{} \frac{f^{(n)}(0)}{n!}x^n$ for $n=0,1,2,3,....$

So, for any ordinary generating function,

sequence is : $a_n= \frac{f^{(n)}(0)}{n!},$ for $n=0,1,2,3,.........$

where, $f^{(n)}$ means $n^{th}$ derivative of $f$.

So, here,

$f(z) =\frac{1+z}{(1-z)^3}= a_{0} z^{0} + a_{1} z^{1} +a_{2} z^{2}+a_3 z^3+........= \sum_{n}^{} \frac{f^{(n)}(0)}{n!}z^n$ for $n=0,1,2,3,....$

So,

$a_0 = \frac{f(0)}{0!}$,

$a_1 = \frac{f'(0)}{1!}$,

$a_2 = \frac{f''(0)}{2!}$,

$a_3 = \frac{f'''(0)}{3!}$ and so on.

Since, here, $f(z) = \frac{1+z}{(1-z)^3}$ $\Rightarrow f(0) = 1$

Now, $f'(z) = \frac{2(z+2)}{(1-z)^4}$

$f''(z) = \frac{6(z+3)}{(1-z)^5}$

$f'''(z) = \frac{24(z+4)}{(1-z)^6}$ $\Rightarrow f'''(0) = 96$

So, $a_0 = \frac{f(0)}{0!} = \frac{1}{0!} = 1$ and $a_3 = \frac{f'''(0)}{3!} = \frac{96}{3!} = 16$

Hence, $a_3 - a_0 = 16 - 1 = 15$

by Boss