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Tagged with binomial-coefficients recurrences
11 questions
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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
$$
W(n, k, m) = (k+m-1)W(n-1,...
- 4,938
5
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3
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How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx) \, dx$$
by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(...
- 513
2
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1
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Find closed-form expression to $f(n)$
For all $n \in \mathbb{N}$, let ${\mathcal A}_n := \left\{\lceil n/2\rceil, \lceil n/2\rceil+1,\dots, n-1 \right\}$ and
$$f(n) := \begin{cases} \min\limits_{a \in {\mathcal A}_n} \frac 1 4 \binom n a ...
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1
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Modulo $2$ binomial transform of A124758
Let $f(n)$ be A153733, remove all trailing ones in binary representation of $n$. Here
\begin{align}
f(2n)& = 2n\\
f(2n+1)& = f(n)\\
\end{align}
Then we have an integer sequence given by
\begin{...
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2
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1
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113
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Modulo $2$ binomial transform of A243499 applied $k$ times
Let $m \geqslant 1$ be a fixed integer.
Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
...
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Inverse modulo $2$ binomial transform of generalised A284005
Let $m \geqslant 1$ be a fixed integer.
Let $\operatorname{wt}(n)$ be A000120, $1$'s-counting sequence: number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $f(n)$ be A007814, ...
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156
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Open tours by a biased rook (proof verification)
Related questions:
Number of open tours by a biased rook on a specific $f(n)\times 1$ board which end on a $k$-th cell from the right
Sum with products turned into subsequences
Combinatorial ...
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2
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400
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Solving a recurrence relation involving binomial coefficients
This question originates from a graph neural network architecture (see 1) in which an edge-labelled graph $G=(V,E,\eta)$ of size $|V|=n$ with $\eta:E\to \mathbb{R}^{s_0}$ is represented as a "tensor''...
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Closed form for unusual recurrence
We have for $k>0$, $n>0$, $m\geqslant0$
$$p_k(n,m)=k(n-1)!\sum\limits_{s=0}^{n-1}\frac{p_{k-1}(s+1,m+1)+p_{k-1}(m+1,s)}{s!}$$
also
$$p_0(n,m)=\begin{cases}
(n-1)!,&\text{$n>0, m=0$}\\
0,&...
- 63.9k
3
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1
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331
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Solving recurrent relation
I have the following recurrent relation and I want to find a close form of it if it exists at all.
$$
P_n = (1-p)^{n-1}P_{n-1} + \sum\limits_{k=2}^{n} \binom{n-1}{k-1} p^{\binom{k}{2}} (1-p)^{k(n-k)} ...
- 60.5k
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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{...
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