Fungrim entry: f4e249

B_{n} = \frac{2 n !}{\pi e} \operatorname{Im}\!\left(\int_{0}^{\pi} {e}^{{e}^{{e}^{i x}}} \sin\!\left(n x\right) \, dx\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1}

References:

https://arxiv.org/abs/0708.3301

TeX:

B_{n} = \frac{2 n !}{\pi e} \operatorname{Im}\!\left(\int_{0}^{\pi} {e}^{{e}^{{e}^{i x}}} \sin\!\left(n x\right) \, dx\right)

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`BellNumber`	$B_{n}$	Bell number
`Factorial`	$n !$	Factorial
`Pi`	$\pi$	The constant pi (3.14...)
`ConstE`	$e$	The constant e (2.718...)
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Sin`	$\sin(z)$	Sine
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("f4e249"),
    Formula(Equal(BellNumber(n), Mul(Div(Mul(2, Factorial(n)), Mul(Pi, ConstE)), Im(Integral(Mul(Pow(ConstE, Pow(ConstE, Pow(ConstE, Mul(ConstI, x)))), Sin(Mul(n, x))), For(x, 0, Pi)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))),
    References("https://arxiv.org/abs/0708.3301"))

Topics using this entry

Bell numbers