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Fungrim entry: 4d2e45

p ⁣(n)=1nk=0n1σ1 ⁣(nk)p ⁣(k)p\!\left(n\right) = \frac{1}{n} \sum_{k=0}^{n - 1} \sigma_{1}\!\left(n - k\right) p\!\left(k\right)
Assumptions:nZ1n \in \mathbb{Z}_{\ge 1}
TeX:
p\!\left(n\right) = \frac{1}{n} \sum_{k=0}^{n - 1} \sigma_{1}\!\left(n - k\right) p\!\left(k\right)

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
PartitionsPp ⁣(n)p\!\left(n\right) Integer partition function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
DivisorSigmaσk ⁣(n)\sigma_{k}\!\left(n\right) Sum of divisors function
ZZGreaterEqualZn\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)), Tuple(k, 0, Sub(n, 1)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

Topics using this entry

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

2019-08-21 11:44:15.926409 UTC