Fungrim entry: fb7a63

p(n) = \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{\infty} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)

Assumptions:

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

TeX:

p(n) = \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{\infty} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p(n)$	Integer partition function
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`HardyRamanujanA`	$A\!\left(n, k\right)$	Exponential sum in the Hardy-Ramanujan-Rademacher formula
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("fb7a63"),
    Formula(Equal(PartitionsP(n), Mul(Div(Mul(2, Pi), Pow(Sub(Mul(24, n), 1), Div(3, 4))), Sum(Mul(Div(HardyRamanujanA(n, k), k), BesselI(Div(3, 2), Mul(Div(Pi, k), Sqrt(Mul(Div(2, 3), Sub(n, Div(1, 24))))))), For(k, 1, Infinity))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

Topics using this entry

Partition function