Exercise · ZW-EXPONENTIAL-GENERATING-FUNCTION-008 · ★★☆ Intermediate
Which sequence \(a_n\) has exponential generating function \(x e^x\)?
Choose one option.
Show answer
- D. \(a_n=n\)
Explanation
Since \(xe^x=\sum_{n\ge 1} x^n/(n-1)!=\sum_n n\,x^n/n!\), we get \(a_n=n\). The option \(a_n=n!\) would give \(\frac{1}{1-x}\).
Review the concept: Exponential generating function
The exponential generating function of a sequence \(a_0, a_1, \ldots\) is the formal power series \(A(x)\) whose coefficient of \(x^n\) is \(a_n/n!\).
\[A(x)=\sum_{n\ge 0}\frac{a_n}{n!}x^n\]
More practice problems: exponential generating function
- ZW-EXPONENTIAL-GENERATING-FUNCTION-001 What is the exponential generating function of the constant sequence a_n=1?
- ZW-EXPONENTIAL-GENERATING-FUNCTION-002 If a_n=n!, what is the exponential generating function?
- ZW-EXPONENTIAL-GENERATING-FUNCTION-003 Which coefficient of x^n defines the exponential generating function of a_n?
- ZW-EXPONENTIAL-GENERATING-FUNCTION-004 What is the series of e^x written as a sum?
- ZW-EXPONENTIAL-GENERATING-FUNCTION-005 Which sequence has exponential generating function e^2x?
- ZW-EXPONENTIAL-GENERATING-FUNCTION-006 Compute a_3 for the sequence whose exponential generating function is (1)/(1-x).
- ZW-EXPONENTIAL-GENERATING-FUNCTION-007 If A(x)=e^3x, what is the coefficient of x^2 in A(x)?
- ZW-EXPONENTIAL-GENERATING-FUNCTION-009 Let A(x)=e^2x and B(x)=e^3x. What is c_4, where C(x)=A(x)B(x) and c_n is the sequence associated wi…
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