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))))

Fungrim entry: dc6806

Topics using this entry