Questions tagged [generating-functions]
Generating functions are formed by making a series $\sum_{n\geq 0} a_n x^n$ out of a sequence $a_n$. They are used to count objects in enumerative combinatorics.
4,700 questions
1 vote
0 answers
31 views
Restricted Factorizations Counting identities
Reading literature related to the problem of multiplicative partitions A001055 OEIS sequence (also named "factorizatio numerorum", "Oppenheim problem", "factorizations ...
- 709
1 vote
2 answers
109 views
Find the number of quadruples of non negative integers $(a,b,c,d)$ such that $105a + 70b + 42c + 30d = 2025$
Find the number of quadruples of non negative integers (a,b,c,d) such that $105a + 70b + 42c + 30d = 2025$ Simply using generating functions makes things very lengthy: This problem is similar to ...
- 579
6 votes
2 answers
458 views
How do I solve this recurrence relation with a squared coefficient?
I started with the following recurrence relation $$ s \left( n \right) = s \left( n-1 \right)^{2} - 1, \qquad s \left( 0 \right) = 2$$ and got to $$\sum_{n=0}^{\infty}s\left(n+1\right)x^{n}=\sum_{n=0}^...
- 83
1 vote
0 answers
133 views
bijective proof of identity coefficient-extracted from negative-exponent Vandermonde identity, and the upper-triangular Stirling transforms
Context: Mircea Dan Rus's 2025 paper Yet another note on notation (a spiritual sequel to Knuth's 1991 paper Two notes on notation) introduces the syntax $x^{\{n\}}=x!{n\brace x}$ to denote the number ...
- 656
0 votes
1 answer
53 views
Extracting coefficients from generating function that is not well-defined in the zero point
In a physics problem I'm currently considering generating functions containing the term $1/\sqrt{t^2}$, where earlier in the derivation I have restricted my attention to the cases $\lvert xt\rvert<\...
- 584
1 vote
1 answer
101 views
Is it possible to solve recurrence relation with non-constant coefficient when coefficient is not explicitly given in terms of $n$?
I have a recurrence relationship as: $$F_{n}=a_nF_{n+1}+a_nF_{n-1}$$ Is it possible to solve such a relation (using a generating function) when the explicit value of $a_n$ is given but not explicitly ...
- 651
0 votes
0 answers
85 views
When does this series converge to a positive integer
I have been working on different unsolved problems of mathematics, mainly for fun and to get a feel for what makes them so difficult. In the process, I stumbled upon this problem, and I am stuck: Let $...
0 votes
0 answers
37 views
Conjectured recurrence for $n$-th row of Salie numbers independently of other rows
Let $B_n$ be the $n$-th Bernoulli number. $T(n,k)$ be A065547, i.e., an integer coefficients known as triangle of Salie numbers whose exponential generating function satisfies $$ \sum\limits_{n=0}^{\...
user929945
1 vote
2 answers
87 views
Exponential generating function for even Eulerian polynomials
Background The Eulerian polynomials $A_{n}(\cdot) $ have the following exponential generating function (e.g.f.): $$ \sum_{n=0}^{\infty} A_{n}(t) \frac{x^{n}}{n!} = \frac{t-1}{t-e^{(t-1)x}} \ . \tag{1}\...
- 7,778
2 votes
3 answers
249 views
Asymptotic approximation of an integer sequence
MOTIVATION I think that computing the asymptotics of $F(x)$ in this answer would give a solution for this question. QUESTION For $n \ge 0$, let $u_{2n} = \binom{2n}{n}^2 / 4^{2n}$ and $u_{2n+1} = 0$. ...
- 2,375
1 vote
1 answer
85 views
New recurrence relation with unusual pattern Is there a closed-form solution? [closed]
I came across an interesting recurrence relation that I couldn’t find any references for: $$ a_1 = 1, \quad a_{n} = \frac{1}{n^2} \sum_{k=1}^{n-1} k \, a_k \, a_{n-k} \quad \text{for } n \geq 2 $$ It ...
- 177
2 votes
0 answers
80 views
Symmetrization of functional derivatives
I'm dealing with a mathematical problem stemming from quantum field theory (QFT). However, at the moment, I'm not concerned with the physics aspect of it and, hence, I wish to view it in purely formal ...
user724377
1 vote
0 answers
61 views
Integer coefficients defined recursively with main diagonal equals A195979
Let $A_n(x)$ be the family of exponential generating fucntions such that $$ A'_n(x) = nA_n(x) + A_{n-1}(x), \\ A_n(0) = 1, A_0(x) = 1. $$ $T(n,k)$ be an integer coefficients whose exponential ...
user929945
2 votes
1 answer
115 views
A combinatorial identity containing double sums: where to find it? [closed]
When I try to establish an explicit formula for computing the generalized Euler polynomials, I encounter with the following identity \begin{equation*} \sum_{m=0}^j\frac{(-1)^m}{2^m}\binom{j}{m}\sum_{...
- 2,306
1 vote
1 answer
60 views
Applying generating function coefficient to find total arrangements
So we have $12$ or fewer balls and $3$ boxes and he has to fill the boxes with the balls with the following constraints - No box is empty Box $B$ has at least $3$ balls & Box $C$ has at most $5$ ...
