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Results tagged with generating-functions
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user 48831
A generating function is a way of encoding an infinite sequence of numbers by treating them as the coefficients of a formal power series. Tag questions involving generating functions in this.
1
vote
Accepted
Asymptotics of coefficients of implicitely defined generating function
Since $a_0\neq 0$ there is (by the (formal) Lagrange theorem) a unique formal power series $x=x(z)$ solving $x={z \over A^3(x)}$, and $A(x(z))=B(z)$ because $A(x)=B(x\,A^3(x))$.
Expanding $A(x)$ using …
- 3,255
2
votes
Accepted
Monotonicity of the sum of coefficients of a family of generating functions
Your conjecture is true. First, observe that
\begin{align*}
K_{w+1}(z)&=(1+z)\,K_w(z) & \mbox{ if } w \mbox{ is odd, }\\
K_{w+1}(z)&=(1+z)\,K_w(z) +\frac{1}{2}{w\choose \frac{w}{2}}z^{w/2}(1-z) & …
- 3,255
1
vote
Accepted
Total progeny of a Galton-Watson branching process - standard textbook question
The answer above is fine, nevertheless I make some hopefully useful supplementary remarks (the first two essentially reformulating Did's answer)
(1) It is well known (see e.g. Feller I, 3rd ed., p.29 …
- 3,255
1
vote
Accepted
Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a C...
The asymptotic distribution of the number of nodes at maximal height in a random tree is known.
The following was known as "Wilf's conjecture" (H. S. Wilf stated it in 1991, evidently unaware of the …
- 3,255