Fungrim entry: a71381

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

Assumptions:

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

References:

https://arxiv.org/abs/0708.3301

TeX:

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

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...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("a71381"),
    Formula(Equal(BellNumber(n), Mul(Div(Mul(2, Factorial(n)), Mul(Pi, ConstE)), Integral(Mul(Mul(Pow(ConstE, Mul(Pow(ConstE, Cos(x)), Cos(Sin(x)))), Sin(Mul(Pow(ConstE, Cos(x)), Sin(Sin(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