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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.

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Let $a(n)$ be an integer sequence with EGF $A(x)$ such that $$ A(x) = \frac{\exp(x)}{\cos(x)-\sin(x)}. $$ $T(n,k)$ be an integer coefficients such that $$ T(n,k) = T(n-1,k) + (k+1)T(n-1,k+1) + kT(n-1,...
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Let $\left(\dots, 0, 0, a_0, a_1, a_2, \dots \right)$ be a totally positive (TP) sequence. Is its corresponding Toeplitz matrix $$A = \begin{bmatrix} a_0 & \cdots & \cdots & \...
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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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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\...
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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) ] } = \...
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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 ...
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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
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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 $$ R(n,k) = qR(n,k-1) + pR(n-1,k-...
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Let $L_n$ denote the set of $n \times n$ lower triangular 0-1 matrices with ones on the diagonal. We associate to $M \in L_n$ a permutation $P$ as the unique permutation in the Bruhat decomposition of ...
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The $\mathcal{P}$-rook polynomial of a polyomino $\mathcal{P}$ is $$ r_\mathcal{P}(T) = \sum_{k=0}^{r(\mathcal{P})} r_k(\mathcal{P})\ T^k, $$ where $r_k(\mathcal{P})$ is the number of ways to place $...
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Let $$ P_{n,d}(q) := \sum_{k=0}^d \binom{n+k-1}{k} q^k $$ denote the Taylor polynomials (of degree $d$) of $\frac{1}{(1-q)^n}$ (truncated binomial series, the coefficients are the multiset ...
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Let $a(n)$ be A318618, i.e., the number of rooted forests on n nodes that avoid the patterns $321$, $2143$ and $3142$ whose exponential function is $A(x)$. Here $$ a(n) = n! \left(1 + \sum\limits_{k=...
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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 ...
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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 ...
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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 ...
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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 ...
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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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Let $T(n,k)$ be A105306 (i.e., triangle read by rows: $T(n,k)$ is the number of directed column-convex polyominoes of area $n$, having the top of the rightmost column at height $k$) whose ordinary ...
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Let $a(n)$ be A000123 (i.e., number of binary partitions: number of partitions of $2n$ into powers of $2$), whose ordinary generating function is $$ \frac{1}{1-x} \prod\limits_{j=0}^{\infty} \frac{1}{...
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Let $T(n,k)$ be an integer coefficients such that $$ T(n,k) = \begin{cases} 1 & \text{if } k = 0 \vee n = k \\ \displaystyle{ \sum\limits_{i=k-1}^{n-1} T(n-1,i) T(i,k-1) } & \text{otherwise} \...
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I recently discovered a recursive, closed-form summation formula that appears to compute the complete homogeneous symmetric polynomial $h_n(x_0, x_1, \dots, x_{m-1})$, but in a more structured and ...
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Let $a(n)$ be A000123, i.e., number of binary partitions: number of partitions of $2n$ into powers of $2$. Here $$ a(2n+1) = a(2n) + a(n), \\ a(2n) = a(2n-1) + a(n), \\ a(0) = 1. $$ $T(n,k)$ be ...
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I have a formal power series $\Phi(q,t)$ that satisfies the following functional equation: $$\Phi(q,t)\cdot \Phi(q^{-1},-t) = 1.$$ Is there a nice known family of functions that satisfies identities ...
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Let $a(n)$ be A007863 whose ordinary generating function satisfies $$ x^2A(x)^3 + xA(x)^2 + (-1+x)A(x) + 1 = 0 $$ and also satisfies $$ A(x) = \exp \left(\sum\limits_{n=1}^{\infty} \left[ \sum\...
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Let $a(n)$ be A382991 whose ordinary generating function is $$ A(x) = 1 + \sum\limits_{i=1}^{\infty} \prod\limits_{j=1}^{i} \left( jx + \frac{x^2}{1-x} \right). $$ $R(n,k)$ be an integer coefficients ...
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Let $T(n,k)$ be A253829 whose ordinary generating function is $$ A(x,z) = \prod\limits_{n=1}^{\infty} \frac{1-z}{1-z-xz^n}. $$ $\nu$ be a vector of fixed length $m$ with elements $\nu_i = \delta_{1,i}...
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Let $f(n)$ and $g(n)$ be an arbitrary functions with integer values defined for $n \geqslant 0$. $T(n,k)$ be an integer coefficients whose ordinary generating function is $$ \cfrac{1}{1-f(0)x-\cfrac{...
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Let $a(n)$ be an arbitrary integer sequence with exponential generating function $A(x)$ and $a(1)=1$. Let $T(n,k)$ be an integer coefficients with exponential generating function $\exp(tA(x))$. Let $b(...
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It is known that the radius of convergence of $A(x)$ is $a$, where $a \in \mathbb{R}$ satisfies $a \geq 1$. The generating function $A(x)$ is: $$ A(x) = \text{Ei}(x) - e^a \text{Ei}(x - a) + \ln\left( ...
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Let $a(n)$ be A103239. Here $$ 1 = \sum\limits_{n=0}^{\infty} a(n) \frac{x^n}{(1-x)^n} \prod\limits_{j=0}^{n} (1 - (j+2)x). $$ Start with $A$ and vector $\nu$ of fixed length $m$ with elements $\nu_i =...
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Let $a(n)$ be an integer sequence with ordinary generating function $A(x)$ such that $$ A(x) = \prod\limits_{k=1}^{\infty} \frac{1}{1+\frac{(-x)^k}{1-x}}. $$ Start with vector $\nu$ of fixed length $m$...
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I'm doing a research and encountered a recurrence relation of the form: $$a_{n+1,s}=\sum_{i=0}^{s}\binom{i+A}{A}(B-s+i)_ia_{n,s-i}$$ where $A$ and $B$ are some integers, and $(m)_k$ is the falling ...
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Here is the 2-dimensional integer lattice reachability problem: Suppose we are in a 2d integer plane, and are starting from $(0, 0)$. We will add $n$ vectors, and each vector is from a pre-defined ...
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Let $a(n)$ be an integer sequence with binary values (that is, $a(n) \in \{0,1\}$) such that there at least one $0$ after each $1$. Let $b(n)$ be an integer sequence of numbers $k$ such that $a(k)=1$. ...
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Let $a(n)$ be an arbitrary integer sequence with exponential generating function $A(x)$ and $a(n)=0$ for $n<1$. Let $T(n,k)$ be an integer coefficients with exponential generating function $\exp(tA(...
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Let $b_0,\,b_1,\,\ldots$ be some given sequence of numbers such that $b_0\neq0$ and $b_0+b_1+\ldots=1$. Let $a_0,\,a_1,\,\ldots$ be some other sequence of numbers whose values are computed as $$ a_0=1,...
Fancier of Mathematica's user avatar
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I have found the hypergeometric identity $$\sum_{k=0}^\infty(22k^2-92k+11)\frac{\binom{4k}k}{16^k}=-5. \tag{1}$$ As the series converges fast, one can easily check $(1)$ numerically by Mathematica. ...
Zhi-Wei Sun's user avatar
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Let $a(n)$ be A001764. Here $$ a(n) = \frac{1}{2n+1}\binom{3n}{n}. $$ Also ordinary generating function is $A(x)$ such that $$ A(x) = 1 + x(A(x))^3. $$ Let $b(n)$ be A251663. Here $$ b(n) = \sum\...
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Let $a(n,q)$ be the family of integer sequences with ordinary generating functions $A_q(x)$ such that $$ A_q(x) = \sum\limits_{n=0}^{\infty} x^n \prod\limits_{k=1}^{n} \frac{(1-(2q(k-1)+1)x)}{(1-(q(2k-...
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Let $a(n,q)$ be the family of integer sequences with ordinary generating functions $A_q(x)$ such that $$ A_q(x) = \sum\limits_{n=0}^{\infty} \frac{x^n}{(1+x)^{n+1}(1-q(n+1)x)}, \\ A_q(x) = \sum\...
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It's well-known (see e.g. the Wikipedia article) that the Lambert $W$ function can be used to solve equations of the form $$x = a e^{b x}$$ by setting $$x = - \frac{W(-ab)}{b}.$$ This follows from the ...
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We have a function $\eta(s,n,m)$ that counts the number of nondecreasing (or nonincreasing) partitions of $s$ with at most $n$ parts, each of size at most $m$. The function can be defined by a ...
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Assume $n$ and $m$ are positive integers. My eventual wish is in finding a $q$-analogue of the below identity but for now I wish to see alternative proofs. I can supply a justification using the Wilf-...
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Let $[c]_q:=\frac{1-q^c}{1-q}=1+q+\cdots+q^{c-1}$ denote the $q$-analogue of the positive integer $c$. For a fixed $r\in\mathbb{N}$, let's write $\vec{c}:=(c_1,\dots,c_r)$ and $\vec{n}:=(n_1,\cdots,...
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I am working with a multivariate normal distribution $\mathbf{x} = [x_1, x_2, \ldots, x_n] \sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, and I need to compute the expectation $E[x_1 x_2 \cdots x_n]...
cloudmath's user avatar
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A friend of mine came out with this riddle: does there exist a nonzero entire function $f(x) = \sum_{k=0}^{+\infty} a_k x^k$ such that for all nonnegative integer $n$, $f(n)=a_n$? I reformulated the ...
Myvh's user avatar
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Consider the generating function for integer partitions $$ P(q)=\frac{1}{\prod_{k=1}^{\infty}(1-q^k)}=\sum_{n=0}^{\infty} p(n)q^n\ , $$ where $p(n)$ is the number of integer partitions of $n$. I am ...
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Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor. $$ Let $T(n,k)$ be an integer coefficients with row length $f(n)$ (number of zeros in the binary expansion of $n$ plus $2$ for $n>0$ with $f(0)=1$...
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Can we find a closed formula for this sum: $$\sum_{p,q\geq 0} (p+q+1)r^{p+q} \frac{{}_1F_1(1+p;2+p+q;r^2)}{{}_1F_1(1+p;2+p+q;1)}$$ where $$_1F_1(a;c;z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n n!} z^...
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Let $\mathbb{N}_+ = \{ 1, 2, \dots\} $. For a given sequence of elements $\{a_i \}_{1 \leq i \leq m} $in $ \mathbb{N}_+ $, we define \begin{equation} P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \...
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