All Questions
11 questions with no upvoted or accepted answers
9
votes
0
answers
258
views
On a continued fraction and vector $\nu$ of length $n$
Please note that this question has been completely reworked in order not to overload it with unnecessary and useless information.
Let $f(n)$ be an arbitrary function with integer values.
Let $a(n)$ ...
- 4,938
6
votes
0
answers
197
views
Where to cut off a double sum?
I have to compute a double infinite sum to within a given accuracy $\epsilon$. Let us say the sum is of the form
$$\sum_{m\geq 1} \sum_{n\geq 1} \frac{a_{m,n}}{m^2 n^2 \max(m,n)},$$
where $|a_{m,n}|\...
- 20.2k
3
votes
0
answers
165
views
Elegant algorithm for A140717
Let $T(n, k)$ be A140717 (i.e., triangle read by rows: $T(n,k)$ is the number of Dyck paths $d$ of semilength $n$ such that sum of peakheights of $d$ - number of peaks of $d$ equals $k$ ($n \geqslant ...
- 4,938
3
votes
0
answers
128
views
Fast and simple algorithm for the A329369
Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\cdots,m\}$ 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 \...
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2
votes
0
answers
182
views
Algorithm for $\frac{1}{1-x} = \sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n}\frac{1-kx}{1+kx}$
Let $a(n)$ be A208832. Here
$$
\frac{1}{1-x} = \sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n}\frac{1-kx}{1+kx}.
$$
Start with vector $\nu$ of fixed length $m$ with elements $\nu_i = 1$ (that ...
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2
votes
0
answers
61
views
Algorithm for main diagonal of integer coefficients associated with Schroeder numbers
Let $T_q(n, k)$ be an integer table such that
$$T_q(n, k) = \begin{cases}
1 & \textrm{if } n = 0 \vee k = 0 \\
qT_q(n-1, n-1) + T_q(n, n-1) & \textrm{if } n = k > 0 \\
T_q(n, k-1) + T_q(n-1,...
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2
votes
0
answers
64
views
On a $\sum\limits_{n=0}^{\infty}c_n x^n=\sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n+1}(1-f(k)x^k)$ (slightly different question)
Please note that this question differs from one of the previous questions of mine.
Let $f(n)$ be an arbitrary function with integer values.
Let $c_n$ be an arbitrary integer sequence.
Let $a(n)$ be ...
- 4,938
2
votes
0
answers
100
views
Another (unique) algorithm for the A329369
Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \leqslant k ...
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1
vote
0
answers
57
views
Step back step forward algorithm for A108442
Let $a(n)$ be A108442. Here generating function is $\frac{1}{1-zA(z)}$ where
$$
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
161
views
Efficient algorithm for A217061
Let $a(n)$ be A217061. Here
$$
a(n) = \sum\limits_{m=1}^{n}\frac{1}{(m-1)!}\sum\limits_{k=0}^{n-m}(n+k-1)!\sum\limits_{j=0}^{k}\frac{1}{(k-j)!}\sum\limits_{\ell=0}^{j}\frac{2^{\ell-j}(-1)^{\ell+j}s(n-...
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0
votes
0
answers
65
views
Algorithm and equivalent recursion for A258173 (related to Dyck paths)
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 $(x,y)$-...
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