Fungrim entry: 4d2e45

p\!\left(n\right) = \frac{1}{n} \sum_{k=0}^{n - 1} \sigma\!\left(n - k\right) p\!\left(k\right)

Assumptions:

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

TeX:

p\!\left(n\right) = \frac{1}{n} \sum_{k=0}^{n - 1} \sigma\!\left(n - k\right) p\!\left(k\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`DivisorSigma`	$\sigma\!\left(n\right)$	Sum of divisors function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("4d2e45"),
    Formula(Equal(PartitionsP(n), Mul(Div(1, n), Sum(Mul(DivisorSigma(Sub(n, k)), PartitionsP(k)), Tuple(k, 0, Sub(n, 1)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

Topics using this entry

Partition function

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

2019-06-18 07:49:59.356594 UTC