Chapter 8: Q21E (page 550)
Give a combinatorial interpretation of the coefficient of in the expansion . Use this interpretation to find this number.
The combinatorial interpretation of the coefficient of is 15.
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Find the coefficient of \({x^{10}}\) in the power series of each of these functions.
a) \({\left( {1 + {x^5} + {x^{10}} + {x^{15}} + \cdots } \right)^3}\)
b) \({\left( {{x^3} + {x^4} + {x^5} + {x^6} + {x^7} + \cdots } \right)^3}\)
c) \(\left( {{x^4} + {x^5} + {x^6}} \right)\left( {{x^3} + {x^4} + {x^5} + {x^6} + {x^7}} \right)(1 + x + \left. {{x^2} + {x^3} + {x^4} + \cdots } \right)\)
d) \(\left( {{x^2} + {x^4} + {x^6} + {x^8} + \cdots } \right)\left( {{x^3} + {x^6} + {x^9} + } \right. \cdots \left( {{x^4} + {x^8} + {x^{12}} + \cdots } \right)\)
e) \(\left( {1 + {x^2} + {x^4} + {x^6} + {x^8} + \cdots } \right)\left( {1 + {x^4} + {x^8} + {x^{12}} + } \right. \cdots )\left( {1 + {x^6} + {x^{12}} + {x^{18}} + \cdots } \right)\)
Find the solutions of the simultaneous system of recurrence relations,
If is the generating function for the sequence, what is the generating function for each of these sequences?
a)
b) (Assuming that terms follow the pattern of all but the first term)
c) (Assuming that terms follow the pattern of all but the first four terms)
d)
e) [Hint: Calculus required here.]
f)
Find the generating function for the finite sequence .
Find a closed form for the generating function for the sequence\(\left\{ {{a_n}} \right\}\), where,
a) \({a_n} = 5\) for all\(n = 0,1,2, \ldots \).
b) \({a_n} = {3^n}\)for all\(n = 0,1,2, \ldots \)
c) \({a_n} = 2\)for\(n = 3,4,5, \ldots \)and\({a_0} = {a_1} = {a_2} = 0\).
d) \({a_n} = 2n + 3\)for all\(n = 0,1,2, \ldots \)
e) \({a_n} = \left( {\begin{array}{*{20}{l}}8\\n\end{array}} \right)\)for all\(n = 0,1,2, \ldots \)
f) \({a_n} = \left( {\begin{array}{*{20}{c}}{n + 4}\\n\end{array}} \right)\)for all\(n = 0,1,2, \ldots \)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
