All Questions
31 questions
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
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
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
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
$$
t\tan\left(\frac{t}{2}\right)=\sum\limits_{n=1}^{\infty}(-1)^n G_{2n}\...
- 4,938
9
votes
2
answers
582
views
Solving a second-order recurrence relation / Series expansion of a confluent Heun equation
I would like to know whether it is possible to solve (in "closed form") either one of the following two second-order recurrence relations, which are closely related to each other. The first ...
- 1,035
0
votes
0
answers
48
views
$R$-recursion for the A007165
Let $a(n)$ be A007165 i.e. number of $P$-graphs with $2n$ edges. Here ordinary generating function $A(x)$ satisfies
$$
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
$$
1, 1, 3, 12, 64, 424, 3358, ...
- 4,938
1
vote
0
answers
89
views
Suitable recursion for the A234289
Let $a(n)$ be A234289 i.e. integer sequence with exponential generating function
$$
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
$$
where the constant of integration is ...
- 4,938
0
votes
0
answers
181
views
Expansion of continued fraction using recursion
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $a(n)$ be an integer sequence with generating function $\frac{1}{G(0)}$ where
$$
G(j)=1-\frac{f(j)x}{G(j+1)}
$$
Here we have
$$
G(...
- 4,938
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
$$b(2^m(2n+1))=\sum\...
- 4,938
14
votes
0
answers
276
views
A conjectured rational generating function
In regard to my question here, let $G_n$
be a sequence of positive integers satisfying
$\lim_{n\to\infty}G_n=\infty$, such that the generating function
$\sum_{n\geq 1} G_nx^n$ is rational. Let
$$ P_n(...
- 50.8k
1
vote
0
answers
118
views
Recurrence relation of the form R(x,y)=yR(x-1,y)+(x-(y-1))R(x,y-1)
Consider the recurrence
$$
R(x,y)= yR(x-1,y)+ (x-(y-1))R(x,y-1)
$$
where for any $R(p,c)$, $c$ does not exceed $p$, and $R(p,p)=R(p,1)=1$.
I´ve tried to write $R(x,y)$ as a sum of coefficients of $R(...
- 11
3
votes
1
answer
1k
views
Special permutations of $\{1,2,3,\ldots,n\}$
How do you show that number of permutations of $\{1,2,3,\ldots,n\}$ such that image of no two consecutive numbers is consecutive is
$$n! + \sum_{k = 1}^{n}(-1)^k\sum_{i = 1}^{k}\dbinom{k - 1}{i - 1}\...
- 149
1
vote
1
answer
591
views
The number of permutations of given order
I want to count the number of permutations of the given order $k$ in $S_n\;(\sigma^k=id,\sigma^l\neq id\;for\;l<k)$. I found some works about that problem, but they are more general than necessary. ...
- 49
0
votes
1
answer
547
views
How to solve this conditional recurrence relation?(two variable and conditions)
I am trying to solve the following recurrence relation
$4 \leq n \;\; \; \; \; \;$ and $\; \; \; \; 2\leq i \leq \lfloor{\frac{n}{2}}\rfloor$
$F(2i,n)=$
$\begin{cases}
\frac{1}{2(2i)-5}F(2i-2,...
- 3
2
votes
1
answer
194
views
Find bivariate generating function for two-dimensional sequence
How to find generating function for triangle of squares of elements in this sequence? I. e. for
$1 + (1 + 4x)y + (1 + 9x + 16x^2)y^2 + ...$ ? It seems that ordinary approach with arithmetic ...
- 195
7
votes
1
answer
394
views
Counting some binary trees with lots of extra stucture
While working on some computations on Hilbert schemes, I came across the following combinatorial problem.
Let $D(k,n)$ be the weighted number of binary trees (children are left/right) with $n$ ...
- 1,509
4
votes
1
answer
518
views
Eliminating a variable from a two-variable linear recurrence
In attempting to enumerate a combinatorial class of objects, I've come to a bivariate recurrence:
$$
a_{n,k} = 2a_{n,k-1} + (k+1)a_{n-1,k+1} - ka_{n-1,k} - a_{n,k-2} + a_{n-1,k-1}.
$$
Together with ...
- 551
1
vote
0
answers
236
views
Solving a recurrence (with the form of a convolution) involving binomial coefficients
While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$
\begin{array}{cccccccccc}
1 & = & K_{1}\tbinom{...
- 11