Seguences ordinary generating functionSolution2 a Let f x

Seguences / ordinary generating function

Solution

2

(a) Let f (x) = 2 + 0x + 0x² +0x + 2x⁴ + 0x⁵ + 0x⁶  + 0x⁷ + 2x⁸ + 0x⁹ + 0x¹⁰+0x¹¹ +2x¹² .... = 2 ( 1+ x⁴+ x⁸ + x¹² +...) = 2* ( 1/ 1-x⁴ ) = 2 / (1 -x⁴) ( the series within the brackets is a geometric series with first term 1 and common ratio x⁴. If mod x⁴ < 1, then the series is convergent and its sum is as indicated). Thus, the generating function is f (x) = 2/ ( 1- x ⁴)

(b) Let g (x) = 0 + 0x +1x²+ 3x³ + 9x⁴ + 27x⁵ + 81x⁶ + 243x⁷+ … = x² ( 1+ 3x +3² x² + 3³ x³ + 3⁴ x⁴ + 3⁵ x⁵ +... = x² ( 1/ 1-3x) = x² / ( 1- 3x) ( the series within the brackets is a geometric series with first term 1 and common ratio 3x. If mod 3x < 1, then the series is convergent and its sum is as indicated). Thus, the generating function is g(x) = x²/ ( 1- 3x )

(x² + x³ )¹⁰ = x ²⁰ ( 1 + x)¹⁰ = x²⁰( x + 1)¹⁰ = x²⁰ ( x¹⁰ + 10x⁹ + 10x⁸ + 10x⁷ + 10x⁶ +... +10x + 1). Thus, the coefficient of x ²⁷ is 10