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Fungrim entry: fb7a63

p(n)=2π(24n1)3/4k=1A ⁣(n,k)kI3/2 ⁣(πk23(n124))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:nZ1n \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
PartitionsPp(n)p(n) Integer partition function
ConstPiπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
HardyRamanujanAA ⁣(n,k)A\!\left(n, k\right) Exponential sum in the Hardy-Ramanujan-Rademacher formula
BesselIIν ⁣(z)I_{\nu}\!\left(z\right) Modified Bessel function of the first kind
Sqrtz\sqrt{z} Principal square root
Infinity\infty Positive infinity
ZZGreaterEqualZn\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, ConstPi), Pow(Sub(Mul(24, n), 1), Div(3, 4))), Sum(Mul(Div(HardyRamanujanA(n, k), k), BesselI(Div(3, 2), Mul(Div(ConstPi, k), Sqrt(Mul(Div(2, 3), Sub(n, Div(1, 24))))))), For(k, 1, Infinity))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC