Fungrim entry: b5a382

B_{n} = {\exp\!\left(\displaystyle{\begin{pmatrix} {0 \choose 0} & {0 \choose 1} & \cdots & {0 \choose N} \\ {1 \choose 0} & {1 \choose 1} & \cdots & {1 \choose N} \\ \vdots & \vdots & \ddots & \vdots \\ {N \choose 0} & {N \choose 1} & \ldots & {N \choose N} \end{pmatrix}} - I_{N + 1}\right)}_{\left(n + 1, 1\right)}

Assumptions:

N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; n \in \{0, 1, \ldots, N\}

TeX:

B_{n} = {\exp\!\left(\displaystyle{\begin{pmatrix} {0 \choose 0} & {0 \choose 1} & \cdots & {0 \choose N} \\ {1 \choose 0} & {1 \choose 1} & \cdots & {1 \choose N} \\ \vdots & \vdots & \ddots & \vdots \\ {N \choose 0} & {N \choose 1} & \ldots & {N \choose N} \end{pmatrix}} - I_{N + 1}\right)}_{\left(n + 1, 1\right)}

N \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; n \in \{0, 1, \ldots, N\}

Definitions:

Fungrim symbol	Notation	Short description
`BellNumber`	$B_{n}$	Bell number
`Exp`	${e}^{z}$	Exponential function
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("b5a382"),
    Formula(Equal(BellNumber(n), Item(Exp(Sub(Matrix(Binomial(i, j), For(i, 0, N), For(j, 0, N)), IdentityMatrix(Add(N, 1)))), Tuple(Add(n, 1), 1)))),
    Variables(N, n),
    Assumptions(And(Element(N, ZZGreaterEqual(0)), Element(n, Range(0, N)))))

Topics using this entry

Bell numbers