Questions tagged [generating-functions]
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.
443 questions
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Invertibility of a semi-infinite lower triangular Toeplitz matrix
Let $\left(\dots, 0, 0, a_0, a_1, a_2, \dots \right)$ be a totally positive (TP) sequence. Is its corresponding Toeplitz matrix
$$A = \begin{bmatrix}
a_0 & \cdots & \cdots & \...
- 7
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Analytic continuation of algebraic functions
Consider the polynomial $K(z,u)= u^e- z (p_0+p_1 u+...+p_k u^k)$, where: $k,e$ are nonnegative integers and the coefficients $p_i$ are nonnegative reals such that $0< \sum_i p_i \leq 1$ with $p_0&...
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Closed form for the convolution type recurrence with coefficients
Let $P_n = P_n(t)$ be the sequence of polynomials such that $P_0=P_1=1$ and it satisfies three equivalent recursive relations:
\begin{align*}
P_{n+1} &= \sum_{k=0}^{n}{n \choose k} \left(kt+k+1\...
- 43
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Expansion identity for the Eulerian polynomials of the second order
Background
$\newcommand{\polylog}{\mathrm{PolyLog}}$
The Eulerian polynomials $A_{m}(\cdot)$ are defined by the exponential generating function:
\begin{equation}
\frac{1-x}{1-x \exp[ t(1-x) ] } = \...
- 5,065
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Similar algorithms for exponential transforms of some exponential generating functions
Let
$a_1(n)$ be A003713, i.e., an integer sequence whose exponential generating function $A_1(x)$ satisfies $$ A_1(x) = \log\left(\frac{1}{1+\log(1-x)}\right). $$
$a_2(n)$ be A141209, i.e., an ...
user231922
7
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370
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A "Lambertization-like" operator on functions
Define an operator $L$ on, say, formal series $f(x)$ with $f(0)=1$ by requiring that $L(f)=F$ is the solution of the functional equation
$$
F(xf(x))=f(x).
$$
Some examples:
\begin{align*}
L(1)&=1;\...
- 19.4k
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Recursion for solution of $(f(x))' = \exp(px) f(qx)$
Let
$a(n)$ be an integer sequence whose exponential generating function $f(x)$ satisfies $$ (f(x))' = \exp(px) f(qx). $$
$R(n,k)$ be an integer coefficients such that $$ R(n,k) = qR(n,k-1) + pR(n-1,k-...
user231922
6
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1
answer
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Counting lower triangular 0-1-matrices with connected Coxeter permutation
Let $L_n$ denote the set of $n \times n$ lower triangular 0-1 matrices with ones on the diagonal. We associate to $M \in L_n$ a permutation $P$ as the unique permutation in the Bruhat decomposition of ...
- 28.1k
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$\mathcal{P}$-rook polynomial of a grid
The $\mathcal{P}$-rook polynomial of a polyomino $\mathcal{P}$ is
$$
r_\mathcal{P}(T) = \sum_{k=0}^{r(\mathcal{P})} r_k(\mathcal{P})\ T^k,
$$ where $r_k(\mathcal{P})$ is the number of ways to place $...
- 1,359
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3
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Are there known explicit closed-form expressions for the Taylor polynomials of $1 / (1-q)^n$?
Let
$$
P_{n,d}(q) := \sum_{k=0}^d \binom{n+k-1}{k} q^k
$$
denote the Taylor polynomials (of degree $d$) of $\frac{1}{(1-q)^n}$ (truncated binomial series, the coefficients are the multiset ...
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Elegant recursion such that its exponential function transform leads to A318618
Let
$a(n)$ be A318618, i.e., the number of rooted forests on n nodes that avoid the patterns $321$, $2143$ and $3142$ whose exponential function is $A(x)$. Here $$ a(n) = n! \left(1 + \sum\limits_{k=...
user231922
2
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1
answer
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Recursion for A014307
Let
$a(n)$ be A014307 whose exponential generating function satisfies $$ A(x) = \sqrt{\frac{\exp(x)}{2-\exp(x)}}. $$
Let $b(n,m)$ be the family of integer sequences whose exponential generating ...
user231922
7
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What really is a $-1$ element set and what does it have to do with the bernoulli numbers?
This is a categorification question:
The sequence of exponential generating functions (indexed by $n$) given by $(e^{x} -1)^n$ allow you to count the number of surjective maps from a $k$ element set ...
- 3,169
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Can a product of non-palindromic $\mathcal{P}$-rook polynomials be palindromic?
I have a question whose answer may be trivial, but I haven't been able to settle it. It concerns the palindromicity of a kind of rook polynomial of a collection of cells.
A polyomino is a set of cells ...
- 1,359
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223
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Is the generating function for triple-turn-avoiding grid Hamiltonian paths D-finite?
Let
$$ \mathcal{G}_{k,n}:=\{1,\dots ,k\}\times\{1,\dots ,n\}\subset\mathbb{Z}^{2}, \qquad k,n\ge1, $$
be a $k\times n$ rectangular lattice graph. A Hamiltonian path (a walk that visits every vertex ...
