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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.

3 votes
0 answers
190 views

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&...
Michele's user avatar
  • 373
3 votes
1 answer
173 views

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\...
linaj's user avatar
  • 33
1 vote
2 answers
296 views

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) ] } = \...
Max Lonysa Muller's user avatar
1 vote
0 answers
79 views

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 ...
user avatar
7 votes
2 answers
363 views

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;\...
მამუკა ჯიბლაძე's user avatar
1 vote
1 answer
165 views

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-...
user avatar
6 votes
1 answer
276 views

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 ...
Mare's user avatar
  • 27.9k
2 votes
0 answers
253 views

$\mathcal{P}$-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 $...
Chess's user avatar
  • 1,309
2 votes
3 answers
580 views

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 ...
M.G.'s user avatar
  • 7,913
1 vote
0 answers
107 views

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=...
user avatar
2 votes
1 answer
225 views

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 ...
user avatar
7 votes
0 answers
288 views

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 ...
Sidharth Ghoshal's user avatar
7 votes
1 answer
551 views

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 ...
Chess's user avatar
  • 1,309
1 vote
1 answer
220 views

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 ...
Alex Cooper's user avatar
1 vote
0 answers
135 views

Is this convolution identity involving Bernoulli numbers known?

I recently discovered the following identity involving Bernoulli numbers: $$ B_{n+m+1} = -\sum_{k=0}^{n} \sum_{v=0}^{m} \frac{(n+m+1)! \, B_k \, B_v}{(n+m-k-v+2)! \, k! \, v!} $$ This holds for all ...
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