All Questions
23 questions with no upvoted or accepted answers
15
votes
0
answers
767
views
Wherefore art thou a Borcherds Product?
This question essentially asks how can one recognize (or rule out) that a generating function of combinatorial origin may be given as a Borcherds type product. I'll start with a motivational example: ...
- 85.6k
7
votes
0
answers
174
views
A diagonal generating function for Fibonacci: Part II
In my earlier MO question, I mentioned although we have for the Fibonacci numbers that
$$F_n=[x^n]\left(\frac1{1-x-x^2}\right),$$
is there a function $F(x)$ such that $F_n=[x^n]\left(F(x)\right)^n$?
...
- 43.2k
5
votes
0
answers
170
views
operation on Ord., Exp., Dri. generating functions
The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by
$$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n\...
- 43.2k
4
votes
0
answers
211
views
Sum $f(n_1,n_2,\ldots,n_k) 1^{n_1} 2^{n_2} \ldots k^{n_k}$ over partitions
Use the notation $(n_1,n_2,\ldots,n_k) \vdash n$ to denote that $(n_1,n_2,\ldots,n_k)$ is a partition of the positive integer $n$, that is, $n_1+n_2+\ldots+n_k = n$ and $n_1 \ge n_2 \ge \ldots \ge n_k ...
- 261
4
votes
0
answers
118
views
Something (which might be called multi-continued fraction) for the A112487
Let $a(n)$ be A112487 i.e. an integer sequence with exponential generating function
$$
A(x)=\exp\left(\int (A(x)+A(x)^2)\,dx\right), \\
A(0)=1
$$
However, the definition in the name of the sequence is
...
- 4,938
4
votes
0
answers
414
views
Explicit formula for tournament sequence
I am looking for an explicit formula for a sequence. The sequence is generated as follows:
There is a tournament with $10$ teams. In the beginning, all teams have a 0-0 win-loss record. The teams are ...
- 41
4
votes
0
answers
312
views
Unexpected result related to open question whether $\sum x^{n^3}$ can satisfy an ADE
In Stanley EC2, it is an open question whether $\sum b_nx^{n^3}$ can satisfy an ADE. Stanley remarks that if this is true then it leads to a "completely unexpected result about representing integers ...
- 220
3
votes
0
answers
70
views
$R$-recursion for the A249833 (similar to A235129)
Let $a(n)$ be A249833 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
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
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
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
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
$$
1, 1, 3, 12, 64, 424, 3358, ...
- 4,938
2
votes
0
answers
110
views
Asking for a generating function for an arithmetic sequence
For fixed integer $n\geq1$, let $c_m(n)$ be the number of divisors $d$ of $m$ such that $n<d\leq 2n$. Here is an experimental generating function for which I ask:
QUESTION. Is this true?
$$\sum_{m\...
- 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
$$
a(n) = \sum\limits_{k=1}^{n}\frac{k}{2n-k}\sum\limits_{i=0}^{n-k} \binom{2n-...
- 4,938
1
vote
0
answers
82
views
Generating functions related to generating function of Catalan numbers
Let $C_n$ be A000108 (i.e., Catalan numbers). Here generating function is $C(x)$ such that
$$
C(x) = \frac{1-\sqrt{1-4x}}{2x}.
$$
Let $a(n)$ be an integer sequence with generating function $A(x)$ such ...
- 4,938
1
vote
0
answers
63
views
On a A162326 and vector $\nu$ of length $n$
Let $a(n)$ be A162326. Here
$$
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}.
$$
Let $b(n)$ be $...
- 4,938
1
vote
0
answers
49
views
$R$-recursion for the A036765
Let $a(n)$ be A036765 i.e. number of ordered rooted trees with $n$ non-root nodes and all outdegrees $\leqslant 3$. Here
$$
a(n) = \frac{1}{n+1}\sum\limits_{j=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\...
- 4,938
1
vote
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
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
$$b(2^m(2n+1))=\sum\...
- 4,938
0
votes
0
answers
48
views
$R$-recursion for the A007165
Let $a(n)$ be A007165 i.e. number of $P$-graphs with $2n$ edges. Here ordinary generating function $A(x)$ satisfies
$$
A(x) = \frac{(1 + xA(x))(1 + 2xA(x))}{1 + 2xA(x) - (xA(x))^2}
$$
Let
$$
R(n, q) = ...
- 4,938
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
0
answers
182
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
$$
G(...
- 4,938
0
votes
0
answers
73
views
Sequences that sum up to possible generalization of Euler or up/down numbers (A000111)
Let $a(n,m,k)$ be an integer sequence with e.g.f.
$$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