Exercise · ZW-EXPONENTIAL-GENERATING-FUNCTION-009 · ★★★ Advanced
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 with \(C\)?
Choose one option.
Show answer
- C. \(5^4=625\)
Explanation
Since \(C(x)=e^{5x}\), the coefficient \(c_4\) is \(5^4=625\). The option with \(\binom{4}{2}\cdot 2^2\cdot 3^2\) confuses the product with a single two-term convolution.
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-008 Which sequence a_n has exponential generating function x e^x?
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