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Results tagged with power-series
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user 48831
7
votes
Comparing two power-series
As in Timothy Budd's answer let $w=w(q)$ denote the (formal) solution of $q=\frac{w}{f(w)}$.
Let $p$ be another variable and consider the sum
\begin{align*}
S(p,q):=\sum_{n,m>0} \sum_{j>0} j[z^{n+j}]\ …
- 3,255
11
votes
What's the summation of formal series $\sum_{n\geq0}\binom{n\delta}{n}x^n$？
The Bürmann-Lagrange theorem gives that
$$\sum_{n\geq 0} {n\delta \choose n} t^n = \frac{1}{1-\delta t(1+z)^{\delta -1}}=\frac{1+z}{1+(1-\delta) z}$$
where $z=z(t)=\sum_{n\geq 1} \frac{1}{n}{n\delta \ …
- 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