# 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

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