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.

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Inverse relationship between Stirling numbers of the first and second kind via generating functions

In combinatorics, a well-known result is that the matrix formed by the Stirling numbers of the second kind $\left(S(n,k)\right)_{n,k\geq 0}$ and the matrix of the signed Stirling numbers of the first ...
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A differential equation and recurrence related to P-partitions

I am interested in polynomials $G_n(z)$ defined by the recurrence $$G_{n+1}(z) - 2G_n(z) + (1-nz)G_{n-1}(z)=0$$ for $n\ge1$ with the initial values $G_0(z) = 1$ and $G_1(z) = 1$. The next few values ...
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5 different ways to define the same family of integer sequences

Let ${n \brace k}$ be a Stirling number of the second kind. Let $A_n(x)$ be an Eulerian polynomial. Here $$A_n(x) = \sum_{i=0}^{n}i!{n \brace i}(x-1)^{n-i}.$$ Let $a_1(n,p,q)$ be the family of ...
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Monotonicity of the sum of coefficients of a family of generating functions

Let \begin{equation*} A_{n,w}(z)=\left(\sum_{i=0}^{\lfloor\frac{w}{2}\rfloor-1}\binom{w}{i}z^i+\frac{1}{2^{(w+1)\bmod 2}}\binom{w}{\lfloor\frac{w}{2}\rfloor}z^{\lfloor\frac{w}{2}\rfloor}\right)^{n/w} \...
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Combinatorial meaning of a binomial expansion

Let $F$ be a generating function $F(x) = \sum_{i=0}^\infty f_i x^i$, and suppose that we can do operations formally without worrying about convergence issues. Define the coefficients \begin{gather*} ...
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Products involving exponents of tribonacci numbers

The Fibonacci numbers $F_n$ can be given by $$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$ Among many many properties of this sequence, consider the following two results: (1) the coefficients of the ...
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Sequences that sum up to possible generalization of Euler or up/down numbers (A000111)

Let $a(n,m,k)$ be an integer sequence with e.g.f. $$A(x)=\operatorname{exp}\left(x + m\int\int (A(x))^k \, dx \, dx\right)$$ I don't know much about integrals, so here's a concrete example: $a(n,1,3)$...
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Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $a(n)$ be an arbitrary integer sequence such that $a(0)=1$. Let $b(n)$ be an integer sequence such that b(2^m(2n+1))=\sum\...