# Fungrim entry: dc6806

$\operatorname{det}\displaystyle{\begin{pmatrix} B_{0 + 0} & B_{0 + 1} & \cdots & B_{0 + n} \\ B_{1 + 0} & B_{1 + 1} & \cdots & B_{1 + n} \\ \vdots & \vdots & \ddots & \vdots \\ B_{n + 0} & B_{n + 1} & \ldots & B_{n + n} \end{pmatrix}} = \prod_{k=1}^{n} k ! = G\!\left(n + 2\right)$
Assumptions:$n \in \mathbb{Z}_{\ge 0}$
TeX:
\operatorname{det}\displaystyle{\begin{pmatrix} B_{0 + 0} & B_{0 + 1} & \cdots & B_{0 + n} \\ B_{1 + 0} & B_{1 + 1} & \cdots & B_{1 + n} \\ \vdots & \vdots & \ddots & \vdots \\ B_{n + 0} & B_{n + 1} & \ldots & B_{n + n} \end{pmatrix}} = \prod_{k=1}^{n} k ! = G\!\left(n + 2\right)

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
BellNumber$B_{n}$ Bell number
Product$\prod_{n} f(n)$ Product
Factorial$n !$ Factorial
BarnesG$G(z)$ Barnes G-function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("dc6806"),
Formula(Equal(Det(Matrix(BellNumber(Add(i, j)), For(i, 0, n), For(j, 0, n))), Product(Factorial(k), For(k, 1, n)), BarnesG(Add(n, 2)))),
Variables(n),
Assumptions(Element(n, ZZGreaterEqual(0))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC