All Questions
Tagged with co.combinatorics recurrences
164 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
67
views
$R$-recursion for A006351
Let $a(n)$ be A006351 (i.e., number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon). Here exponential generating function is $A(x)$ such that $B(x) = ...
- 4,938
2
votes
0
answers
59
views
$R$-recursion for A338193
Let $a(n)$ be A338193. Here generating function is $A(x)$ such that
$$
A(x) = 1 + \int\frac{\left(\frac{x}{A(x)}\right)'}{\left(\frac{x}{(A(x))^2}\right)'} \, dx.
$$
Let
$$
R(n, q) = \begin{cases}
1 &...
- 4,938
1
vote
1
answer
92
views
Equivalence of sequences related to A033264
Let $a(n)$ be A033264 (i.e., number of blocks of $\{1,0\}$ in the binary expansion of $n$). Here
$$
a(4n) = a(4n+1) = a(2n), \\
a(4n+2) = a(n)+1, \\
a(4n+3) = a(n), \\
a(0) = 0.
$$
Let
$$
\ell(n) = \...
- 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
46
views
On A088352 as an antidiagonal sums of A129179
Let $a(n)$ be A088352. Here $a(n)$ is an integer sequence with generating function $A(x)$ such that
$$
A(x) = \cfrac{1}{1-x-\cfrac{x^2}{1-x^3-\cfrac{x^4}{1-x^5-\cfrac{x^6}{1-x^7-\cfrac{x^8}{\ddots}}}}}...
- 4,938
1
vote
0
answers
32
views
On a A347205 and related row polynomials
Let $a(n)$ be A347205. Here
$$
a(2^m(2k+1)) = \sum\limits_{j=0}^{m}a(2^j k), \\
a(0) = 1.
$$
Let $\nu_2(n)$ be A007814 (i.e., number of trailing zeros in the binary expansion of $n$). Here
$$
\nu_2(2n+...
- 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 ...
0
votes
0
answers
73
views
General solution of partial difference equation that generates Eulerian numbers
I have a question on the partial difference equation
$$f(n+1, k) = (k+1) f(n,k) + (n+1-k)f(n,k-1)$$
where $(k, n) \in \mathbb{Z}^2$.
It is well known, that under some boundary conditions this equation ...
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
0
votes
0
answers
135
views
Integer coefficients and continued fractions
Let $a(n,p,q)$ be the family of integer sequences such that ordinary generating functions for it are $\frac{1}{G_1(0,x)}$ where $G_1(0,x)$ are continued fractions such that
$$
G_1(j,x)=1-\cfrac{(qj+1)...
- 4,938
3
votes
1
answer
140
views
$R$-recursion for unsigned Genocchi numbers (of first kind) of even index
Let $G_n$ be A036968 (i.e., Genocchi numbers). Here
$$
\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
1
vote
1
answer
61
views
Simplest way to generate integer coefficients with row sums equal to the terms of an arbitrary given sequence
Let $f(n)$ be an arbitrary function.
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\operatorname{wt}(2n+1) = \operatorname{wt}(n) + 1, \\
\...
- 4,938
1
vote
0
answers
168
views
Integer coefficients and integrals
Let $a(n,p,q)$ be the family of integer sequences such that exponential generating functions for it satisfy
$$
A_1(x)=\exp\left(x + p\int\int (A_1(x))^q \, dx \, dx\right).
$$
Let $b(n,p,q)$ be the ...
- 4,938
0
votes
0
answers
55
views
Sequences that sum up to sums of integer coefficients
Let
$$
T(n,k,p,q,r,s) = (q(k-1)+1)T(n-1,k,p,q,r,s) + s(n+r(k-1)+p-2)T(n-1,k-1,p,q,r,s), \\
T(n,1,p,q,r,s) = 1, \\
T(n,0,p,q,r,s) = T(0,k,p,q,r,s) = 0
$$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\...
- 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
8
votes
0
answers
150
views
Can P-recursive functions assume only prime values?
A function $f\colon \{0,1,\dots\}\to \mathbb{R}$ is P-recursive if
it satisfies a recurrence $$
P_d(n)f(n+d)+P_{d-1}(n)f(n+d-1)+\cdots+P_0(n)f(n)=0,\ n\geq 0, $$
where each $P_i(n)\in \mathbb{R}[n]$ ...
- 50.8k
1
vote
0
answers
90
views
Closed form for the A357990 using A329369 and generalised A373183
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor, \\
\ell(0) = -1
$$
Let
$$
f(n) = \ell(n) - \ell(n-2^{\ell(n)}) - 1
$$
Here $f(n)$ is A290255.
Let $A(n,k)$ be a square array such that
$$
A(n,k)...
- 4,938
3
votes
0
answers
129
views
Sequence that sums up to A014307
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $n \brace k$ be a Stirling number of the second kind.
Let $a(n)$ be A014307. Here
$$
A(x) = \sum\limits_{k=0}^{\infty} \frac{a(k)}{...
- 4,938
2
votes
1
answer
129
views
Recursion for the sum with Stirling numbers of both kinds
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $n \brace k$ be a Stirling number of the second kind.
Let
$$
f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace j}...
- 4,938
2
votes
0
answers
100
views
Another (unique) algorithm for the A329369
Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \leqslant k ...
- 4,938
1
vote
0
answers
58
views
Simple recursion for the A329369 using Stirling numbers of both kinds
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let $n \brace k$ be a Stirling number of the second kind.
Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with ...
- 4,938
1
vote
0
answers
132
views
Sequence that sums up to A000153
Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \leqslant k ...
- 4,938
3
votes
0
answers
88
views
Recursion for reversed rows of the A373183 using unsigned Stirling numbers of the first kind
Let $\left[{n \atop k}\right]$ be unsigned Stirling numbers of the first kind. Here
$$
\left[{n \atop k}\right] = (n-1)\left[{n-1 \atop k}\right] + \left[{n-1 \atop k-1}\right], \\
\left[{n \atop 0}\...
- 4,938
1
vote
0
answers
105
views
Simpler recursion for the A358612
Let $T(n,k)$ be an integer coefficients (A358612) such that
$$
T(2n+1, k) = kT(n, k) + T(n, k-1), \\
T(2n, k) = kT(n, k) + T(n, k-1) - \frac{T(2n, k-1) + T(n, k-1)}{k-1}, \\
T(n, 1) = T(0, 2) = 1
$$
...
- 4,938
2
votes
0
answers
70
views
Property of a family of simple polynomials related to the A329369
Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\dotsc,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...
- 4,938
1
vote
0
answers
73
views
Alternating sum of integer coefficients of the triangles related to Eulerian numbers and binomial transforms
Let $W(n, k, m)$ be an integer coefficients defined for $n > 0, 1 \leqslant k \leqslant n, m > 0$ with $W(n,k,m)=0$ for $n \leqslant 0$ or $k \leqslant 0$ such that
$$
W(n, k, m) = (k+m-1)W(n-1,...
- 4,938
6
votes
0
answers
245
views
Searching for a proof of the pattern and identification of integer coefficients for the A329369
Please see the update given below. Everything you need to know from the old version of the question are the functions $a(n), \ell(n), s(n), t(n), r(n)$.
Let $a(n)$ be A329369 (i.e, number of ...
- 4,938
6
votes
1
answer
367
views
On A057985 and A287066
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0 \to 01$, $1 \to 12$, $2 \to 0$).
Let $b(n)$ be A287066 (i.e., start with $1$ and repeatedly substitute: $0 \to 01$, $1 \to 12$...
- 4,938
2
votes
1
answer
252
views
On a A089039 and pair of sequences with simple recursion
Let $a(n)$ be A089039 (i.e., number of circular permutations of $2n$ letters that are free of jealousy). Here
$$
a(n) = \sum\limits_{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\frac{n!(n-k-1)!^2}{(k-...
- 4,938
2
votes
1
answer
120
views
Recursion for the Chebyshev transform of $m^n$
Let
$$
R(n, q, m) = R(n-1, q+1, m) + \sum\limits_{j=0}^{q} (-1)^{q-j}R(n-1, j, m), \\
R(0, q, m) = (m-1)^q
$$
I conjecture that $R(n, 0, m)$ is a Chebyshev transform of $m^n$.
Examples of Chebyshev ...
- 4,938
1
vote
1
answer
114
views
Closed form for a linear recurrence relation of varying order
In my research I have come across a recurrence relation that is of varying order. The relation is as follows:
$$
\begin{cases}
f_0=f_1=0,\\
f_2=1,\\
\bigg(f_{2\rho}=\displaystyle \sum_{i=0}^{\rho}...
1
vote
0
answers
68
views
On a numbers $k$ with specific $2$-adic valuation
Let $a(n)$ be A002326 (i.e., multiplicative order of $2 \operatorname{mod} 2n+1$).
Let $b(n)$ be A179382 (i.e., the smallest period of pseudo-arithmetic progression with initial term $1$ and ...
- 4,938
0
votes
0
answers
48
views
$R$-recursion for the A007165
Let $a(n)$ be A007165 i.e. number of $P$-graphs with $2n$ edges. Here ordinary generating function $A(x)$ satisfies
$$
A(x) = \frac{(1 + xA(x))(1 + 2xA(x))}{1 + 2xA(x) - (xA(x))^2}
$$
Let
$$
R(n, q) = ...
- 4,938
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
$$
a(n) = \frac{1}{n+1}\sum\limits_{j=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\...
- 4,938
2
votes
1
answer
167
views
$R$-recursion for the A143017
Let $a(n)$ be A143017 i.e. number of $\{2-1-3, 2'^e-31\}$-avoiding permutations of size $n$ (see definition in the Elizalde paper). Here
$$
a(n) = \frac{1}{n}\sum\limits_{k=0}^{\left\lfloor\frac{n}{...
- 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
$$
A(x)=\sum\limits_{i=0}^{\infty}\frac{x^i}{\prod\limits_{j=0}^{i}(1-f(j)x)...
- 4,938
2
votes
0
answers
69
views
Set partitions with big blocks - real-rooted polynomials?
The polynomials
$$
T_n(t) := \sum_{\pi \in \text{Set Partitions}(n)} t^{\text{blocks}(\pi)} = \sum_{k=1}^n S(n,k)t^k
$$
with $S(n,k)$ being the Stirling numbers of the second kind, are well-known to ...
- 15.8k
6
votes
0
answers
198
views
Filling in some missing squares for classes of power series
This question concerns various important classes of formal power series. For concreteness and convenience, let us work with power series $F(x) = \sum_{n\geq 0}c_n x^n \in \mathbb{C}[[x]]$, i.e., with ...
- 24.2k
1
vote
1
answer
99
views
$R$-recursion for the A307389
Let $a(n)$ be A307389 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
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
2
votes
0
answers
70
views
Elementary recursion for the A258173
Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$.
A Dyck path of semilength $n$ is a $(x,y)$-...
- 4,938
2
votes
0
answers
140
views
Asymptotics of a "non-constant order" quadratic recurrence relation in two variables
Consider the following recurrence relation defined for two integer variables $H,n \geq 0$:
\begin{equation}
\gamma(H,n) = \sum_{K=0}^{\lfloor H/2 \rfloor} \gamma(K,n-1) \gamma(H-K,n-1)
\end{equation}
...
- 143
2
votes
0
answers
90
views
Unexpected recursion for the A193231 (blue code of $n$)
Let $a(n)$ be A193231, blue code of $n$ i.e. self-inverse permutation of non-negative integers such that $a(n)<2^k$ iff $n<2^k$ and
$$
a(n\operatorname{XOR}k) = a(n) \operatorname{XOR} a(k)
$$
...
- 4,938
1
vote
1
answer
108
views
Property of some permutations of non-negative integers such that $a(n)<2^k$ iff $n<2^k$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
Let
$$
f(n) = 2^{\ell(n)}
$$
Let $q_1(n)$ and $q_2(n)$ be an arbitrary self-inverse permutations of non-negative integers (that is, $q_i(q_i(n)) ...
- 4,938
3
votes
0
answers
121
views
Closed form from a slightly modified recursion for transposed Catalan triangle
Let $a_1(n)$ be A000108, i.e. Catalan numbers. Here
$$
a_1(n)=\frac{1}{n+1}\binom{2n}{n}
$$
Let $a_2(n)$ be A059715, i.e. number of multi-directed animals on the triangular lattice. From OEIS page we ...
- 4,938