# 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

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