All Questions
Tagged with co.combinatorics generating-functions
227 questions
0
votes
1
answer
49
views
Formula for $P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \mathbb{N}_+} \left( \prod_{i=1}^m k_i^{a_i} \right) $
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 \...
- 89
4
votes
1
answer
197
views
Solving a three-parameter recursive sequence
Consider the triple-indexed sequence of integers defined by
\begin{align} \label{coefficientsV} \nonumber
f(\alpha,\beta,\gamma)
&:=(2\alpha+8\beta+12\gamma-1)\cdot f(\alpha-1,\beta,\gamma)...
- 43.2k
1
vote
0
answers
57
views
Step back step forward algorithm for A108442
Let $a(n)$ be A108442. Here generating function is $\frac{1}{1-zA(z)}$ where
$$
A(z) = 1 + z(A(z))^2 + z(A(z))^3.
$$
Also
$$
a(n) = \sum\limits_{k=1}^{n}\frac{k}{2n-k}\sum\limits_{i=0}^{n-k} \binom{2n-...
- 4,938
1
vote
1
answer
77
views
Enumeration of permutations with prescribed numbers of fixed points and excedance/deficiency statistics
Consider the following refinement of permutation statistics. For $π ∈ S_n$, let:
$\mathrm{fix}(π) = |\{i : π(i) = i\}|$ (number of fixed points)
$\mathrm{exc}(π) = |\{i : π(i) > i\}|$ (number of ...
- 11
1
vote
0
answers
47
views
Harmonic numbers multifold convolution
I have a question. If I define the multifold convolution of Harmonic numbers as
$\sum_{n_1=1}^{\infty} \cdots \sum_{n_k=1}^{\infty} H_{n_1} \cdots H_{n_k} \mathbf{1}_{\{n\}}(n_1+\dots+n_k)$
for the $k$...
- 471
1
vote
0
answers
82
views
Generating functions related to generating function of Catalan numbers
Let $C_n$ be A000108 (i.e., Catalan numbers). Here generating function is $C(x)$ such that
$$
C(x) = \frac{1-\sqrt{1-4x}}{2x}.
$$
Let $a(n)$ be an integer sequence with generating function $A(x)$ such ...
- 4,938
2
votes
0
answers
51
views
Recursion for A129179 similar to recursion for Pascal's triangle
Let $T(n,k)$ be A129179 (i.e., triangle read by rows: $T(n, k)$ is the number of Schroeder paths of semilength $n$ such that the area between the $x$-axis and the path is $k$ ($n \geqslant 0, 0 \...
- 4,938
2
votes
1
answer
310
views
Generating function for A300483 (related to Chebyshev polynomial of first kind)
Let $a(n)$ be A300483. Here
$$
a(n) = 2\int\limits_{t \geqslant 0}T_n\left(\frac{t+1}{2}\right)\exp(-t)\,dt.
$$
where $T_n(x)$ is $n$-th Chebyshev polynomial of first kind.
Let $b(n)$ be an integer ...
- 4,938
10
votes
1
answer
625
views
Generating function for A261041
Let $a(n)$ be A261041 (i.e., number of partitions of subsets of $\{1,2,\dotsc,n\}$, where consecutive integers are required to be in different parts).
Let $b(n)$ be an integer sequence with generating ...
- 4,938
10
votes
1
answer
207
views
Generating function for A225114
Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns).
Let $b(n)$ be an integer sequence with generating function $B(x)$ such that
$$
B(x) = \...
- 4,938
2
votes
0
answers
64
views
On a $\sum\limits_{n=0}^{\infty}c_n x^n=\sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n+1}(1-f(k)x^k)$ (slightly different question)
Please note that this question differs from one of the previous questions of mine.
Let $f(n)$ be an arbitrary function with integer values.
Let $c_n$ be an arbitrary integer sequence.
Let $a(n)$ be ...
- 4,938
7
votes
3
answers
707
views
Properties of $P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$
I know this will sound like a general question, but given the structure $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what ...
5
votes
1
answer
168
views
On a generating function and vector $\nu$ of length $n$
Let $f(n)$ be an arbitrary function with integer values.
Let $a(n)$ be an integer sequence such that
$$
\frac{1}{1-x}=\sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n+1}(1-f(k)x)
$$
Start with ...
- 4,938
1
vote
0
answers
63
views
On a A162326 and vector $\nu$ of length $n$
Let $a(n)$ be A162326. Here
$$
a(n) = \frac{1}{n}(2(5n-7)a(n-1) - 9(n-2)a(n-2)), \\
a(0) = a(1) = 1.
$$
Also ordinary generating function is
$$
\frac{5 - \sqrt{\frac{1-9x}{1-x}}}{4}.
$$
Let $b(n)$ be $...
- 4,938
4
votes
1
answer
406
views
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 ...
- 151
7
votes
0
answers
162
views
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 ...
- 17k
2
votes
2
answers
315
views
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 ...
- 4,938
3
votes
1
answer
140
views
$R$-recursion for unsigned Genocchi numbers (of first kind) of even index
Let $G_n$ be A036968 (i.e., Genocchi numbers). Here
$$
\frac{2t}{1+e^t}=\sum\limits_{n=0}^{\infty}G_n\frac{t^n}{n!}.
$$
Also
$$
t\tan\left(\frac{t}{2}\right)=\sum\limits_{n=1}^{\infty}(-1)^n G_{2n}\...
- 4,938
9
votes
2
answers
582
views
Solving a second-order recurrence relation / Series expansion of a confluent Heun equation
I would like to know whether it is possible to solve (in "closed form") either one of the following two second-order recurrence relations, which are closely related to each other. The first ...
- 1,035
2
votes
2
answers
205
views
Monotonicity of the sum of coefficients of a family of generating functions
Let
\begin{equation*}
A_{n,w}(z)=\left(\sum_{i=0}^{\lfloor\frac{w}{2}\rfloor-1}\binom{w}{i}z^i+\frac{1}{2^{(w+1)\bmod 2}}\binom{w}{\lfloor\frac{w}{2}\rfloor}z^{\lfloor\frac{w}{2}\rfloor}\right)^{n/w}
\...
- 23
5
votes
1
answer
231
views
Reference request: Gessel interview's generating function identities
In this interview, Ira Gessel mentions the following results:
Result 1: Let $B_n$ denote the $n^{\text{th}}$ Bernoulli number.
Define the series
$$B(x) = \sum_{n=2}^{\infty} \frac{B_nx^{n-1}}{n(n-1)}.$...
- 557
2
votes
1
answer
196
views
Weak compositions with no subcomposition adding to (more than) $j$
Here is a solution to a problem from Stanley's Enumerative Combinatorics (it's listed as a difficulty 2, so I imagine what I'm about to ask is likely a 2+ or 3-) about the number $\kappa(N,k,j)$ of ...
- 23
9
votes
2
answers
344
views
Counting $m\times n$ $\bigl({1\atop1}{1\atop0}\bigr)$-free $(0,1)$-matrices
Let $G_{m,n}$ denote the number of $m\times n$ $(0,1)$-matrices that avoid the submatrix $\bigl({1\atop1}{1\atop0}\bigr)$. (Submatrices need not be contiguous.) Here are some small values (not yet on ...
- 667
5
votes
1
answer
148
views
Identities for the generating functions of a sort of convolution powers of the Narayana numbers
Let $c(x)=\frac{1-\sqrt{1-4x}}{2x}$ be the generating function of the Catalan numbers.
It satisfies $$\frac{1}{c(x)^k}+x^k c(x)^k=L_k(1,-x),$$
where $L_n(x,s)$ denote the Lucas polynomials defined by $...
- 5,622
2
votes
1
answer
259
views
The probability that iid draws from a mean zero random variable sum to zero
Suppose we have a probability distribution $p(\cdot)$ supported on the integers between $-m$ and $m$ for some positive integer $m$, with $\sum_k kp(k) = 0$. Suppose furthermore that all $p(k)$ are ...
- 19.7k
0
votes
0
answers
48
views
$R$-recursion for the A007165
Let $a(n)$ be A007165 i.e. number of $P$-graphs with $2n$ edges. Here ordinary generating function $A(x)$ satisfies
$$
A(x) = \frac{(1 + xA(x))(1 + 2xA(x))}{1 + 2xA(x) - (xA(x))^2}
$$
Let
$$
R(n, q) = ...
- 4,938
1
vote
0
answers
49
views
$R$-recursion for the A036765
Let $a(n)$ be A036765 i.e. number of ordered rooted trees with $n$ non-root nodes and all outdegrees $\leqslant 3$. Here
$$
a(n) = \frac{1}{n+1}\sum\limits_{j=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\...
- 4,938
1
vote
1
answer
116
views
General case of the some $R$-recursions
Let $f(n)$ be an arbitrary function.
Let $a(n)$ be an integer sequence such that its ordinary generating function satisfies
$$
A(x)=\sum\limits_{i=0}^{\infty}\frac{x^i}{\prod\limits_{j=0}^{i}(1-f(j)x)...
- 4,938
1
vote
1
answer
99
views
$R$-recursion for the A307389
Let $a(n)$ be A307389 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A(x)=\exp\left(\frac{\exp(2x)-2\exp(x)+2x+1}{2}\right)
$$
The sequence begins with
$$
1,...
- 4,938
3
votes
0
answers
70
views
$R$-recursion for the A249833 (similar to A235129)
Let $a(n)$ be A249833 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A(x) = 1 + \int A(x) + (A(x))^2\log A(x)\,dx
$$
The sequence begins with
$$
1, 1, 2, 7, ...
- 4,938
2
votes
0
answers
103
views
$R$-recursion for the A235129
Let $a(n)$ be A235129 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A'(x) = 1 + A(x)\exp(A(x))
$$
The sequence begins with
$$
1, 1, 3, 12, 64, 424, 3358, ...
- 4,938
20
votes
2
answers
741
views
A rational function related to Fibonacci numbers
Let $F_n$ denote a Fibonacci number ($F_1=F_2=1$,
$F_{n+1}=F_n+F_{n-1}$ for $n\geq 2$). Define
$$\prod_{k=1}^n (1+x^{F_{k+1}}) = \sum_j f(n,j)x^j. $$
For a positive integer $r$ let
$$ v_r(n) = \sum_j ...
- 50.8k
7
votes
0
answers
279
views
A conjecture about Hankel determinants of path generating functions
Let $a_{n,k}=a_{n,k}(x,c)$ be the generating function $\sum_P w(P),$ where $P$ runs over all paths from $(0,0)$ to $(n,k)$ consisting of horizontal steps $(1,0)$, up-steps $(1,1)$ and down-steps $(1,-...
- 5,622
7
votes
1
answer
503
views
Combinatorial consequences of de Branges's Theorem?
I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
- 191
3
votes
0
answers
222
views
Number of partitions of set restricted by sum of square of part size
Let $p_1^{a_1}p_2^{a_2}\cdots$ denotes the integer partition of $n$, i.e. $a_1p_1+a_2p_2+\cdots=n$. Or equivalently $m_1+m_2+\cdots=n$. It is known that the number of partitions of set $\{x_1,x_2,\...
- 405
2
votes
1
answer
335
views
Combinatorial meaning of a binomial expansion
Let $F$ be a generating function $F(x) = \sum_{i=0}^\infty f_i x^i$, and
suppose that we can do operations formally without worrying about
convergence issues.
Define the coefficients
\begin{gather*}
...
- 5,230
4
votes
0
answers
211
views
Sum $f(n_1,n_2,\ldots,n_k) 1^{n_1} 2^{n_2} \ldots k^{n_k}$ over partitions
Use the notation $(n_1,n_2,\ldots,n_k) \vdash n$ to denote that $(n_1,n_2,\ldots,n_k)$ is a partition of the positive integer $n$, that is, $n_1+n_2+\ldots+n_k = n$ and $n_1 \ge n_2 \ge \ldots \ge n_k ...
- 261
3
votes
1
answer
156
views
$q$-series and Stirling of the 1st kind
Denote the (unsigned) Stirling numbers of the $1^{st}$-kind by ${n \brack k}$ and define
$$\mathbf{F}_a(q)=\sum_{m\geq1}\frac{q^{am}}{(1-q^m)^{2a}} \qquad \text{and} \qquad
\mathbf{G}_b(q)=\sum_{m\...
- 43.2k
1
vote
0
answers
89
views
Suitable recursion for the A234289
Let $a(n)$ be A234289 i.e. integer sequence with exponential generating function
$$
A(x)=1+A(x)^2\int \frac{1}{A(x)}\,dx
$$
The sequence begins with
$$
1, 1, 3, 17, 147, 1729, 25827, 468593, 10012083, ...
- 4,938
1
vote
0
answers
80
views
Recursion for the A006014 using difference of binomial coefficients
Let $a(n)$ be A006014 i.e.
$$
a(n)=na(n-1)+\sum\limits_{j=1}^{n-2}a(j)a(n-j-1), \\
a(1)=1
$$
Also generating function $A(x)$ satisfies
$$
A(x) = x(1 + A(x) + A(x)^2 + xA'(x))
$$
Let
$$
R(n,q)=\sum\...
- 4,938
4
votes
0
answers
118
views
Something (which might be called multi-continued fraction) for the A112487
Let $a(n)$ be A112487 i.e. an integer sequence with exponential generating function
$$
A(x)=\exp\left(\int (A(x)+A(x)^2)\,dx\right), \\
A(0)=1
$$
However, the definition in the name of the sequence is
...
- 4,938
0
votes
0
answers
100
views
Recursion for the A266328 by analogy with non-standard recursion for factorials
Let $a(n)$ be A266328 i.e. an integer sequence with exponential generating function
$$
A(x)=\exp\int B(x) \,dx
$$
such that
$$
B(x)=\exp(-x)\exp\int A(x) \,dx
$$
where the constant of integration is ...
- 4,938
0
votes
2
answers
331
views
Simplification of hypergeometric Function
First of all I am not at all a math expert, but I have some working knowledge.
That said, please excuse "dumb" questions.
I am looking at the following process: Assume you are on the 2-...
- 31
0
votes
0
answers
181
views
Expansion of continued fraction using recursion
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $a(n)$ be an integer sequence with generating function $\frac{1}{G(0)}$ where
$$
G(j)=1-\frac{f(j)x}{G(j+1)}
$$
Here we have
$$
G(...
- 4,938
4
votes
0
answers
208
views
Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
1
vote
1
answer
344
views
Products involving exponents of tribonacci numbers
The Fibonacci numbers $F_n$ can be given by
$$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$
Among many many properties of this sequence, consider the following two results:
(1) the coefficients of the ...
- 43.2k
0
votes
0
answers
73
views
Sequences that sum up to possible generalization of Euler or up/down numbers (A000111)
Let $a(n,m,k)$ be an integer sequence with e.g.f.
$$A(x)=\operatorname{exp}\left(x + m\int\int (A(x))^k \, dx \, dx\right)$$
I don't know much about integrals, so here's a concrete example:
$a(n,1,3)$...
- 4,938
1
vote
0
answers
93
views
Application of the series reversion
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $a(n)$ be an arbitrary integer sequence such that $a(0)=1$.
Let $b(n)$ be an integer sequence such that
$$b(2^m(2n+1))=\sum\...
- 4,938
0
votes
1
answer
195
views
Fibonacci and product polynomials
The motivation for my current question arises from this MO post by R. Stanley. Caveat. There's a slight alteration.
With the convention $F_1=F_2=1$ for the Fibonacci numbers, define the polynomials $...
- 43.2k
8
votes
1
answer
275
views
Use of generating functions in logic
Are there any uses of generating functions within logic, in particular to count how many models exists for a given theory $T$, say in FOL?
The concrete problem I'm hoping to apply this to involves ...
- 193