Fungrim entry: 4d2e45

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p(n)$	Integer partition function
`Sum`	$\sum_{n} f(n)$	Sum
`DivisorSigma`	$\sigma_{k}\!\left(n\right)$	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(1, Sub(n, k)), PartitionsP(k)), For(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.

2021-03-15 19:12:00.328586 UTC