All Questions
Tagged with integer-partitions sequences-and-series
45
questions
6
votes
0
answers
323
views
Proof of these identities
The following is a screenshot from a paper by Daniel B. Grunberg called On asymptotics, Stirling numbers, Gamma function, and polylogs.
I only offer the page as a reference to explain equation 3.1. ...
- 3,557
1
vote
1
answer
189
views
Index set of dyadic partition
Imagine if you have a line from 0 to 1, and you begin partitioning it dyadically. The first point will be at 0, the second at 1 and the third at 0.5, the fourth at 0.25, the fifth at 0.75 etc. Let's ...
- 11
1
vote
0
answers
43
views
How to derive the proof for the "intermediate series" in the Euler Transform?
Consider the description in Wolfram Alpha of a "third kind" of Euler Transform (http://mathworld.wolfram.com/EulerTransform.html), expressions (5), (7), and (8):
(5) $1+\sum_{n=1}^\infty b_nx^n=\...
- 550
2
votes
1
answer
1k
views
Formula for how many combinations of powers of 2 sum to $2^n$
Given a number $2^n, n\in\mathbb{Z}\gt 0$, I would like to find a formula for how many unique sets of powers of $2$ sum to that number. This is related to the triangular numbers but excludes non-...
- 1,096
9
votes
0
answers
164
views
Closed form for $\sum_{n=1}^\infty \frac{1}{P(n)}$, where $P(n)$ is the partition function.
Is there a closed form for the following infinite series?
$$\sum_{n=1}^\infty \frac{1}{P(n)}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$$
where $P(n)$ is the partition function.
- 1,893
1
vote
1
answer
55
views
What is the name of the transform which finds the number of ways to make partitions of the given sizes?
I'm looking for the name of a transform which takes a sequence giving the number of 'prime' elements of a given size to the number of ways to make a number out of a sum of 'prime' elements, up to ...
- 32.3k
8
votes
1
answer
1k
views
Elementary proof of Ramanujan's "most beautiful identity"
Ramanujan presented many identities, Hardy chose one
which for him represented the best of Ramanujan. There are many proofs for this identity.
(for example, H. H. Chan’s proof, M. Hirschhorn's proof....
- 1,893
3
votes
2
answers
144
views
how interpret this partition identity?
use the symbol $P(N)$ to denote the set of all partitions of a positive integer $N$ and denote by $P_k$ the number of occurrences of $k$ in the partition $P \in P(N)$, so that
$$
N = \sum kP_k
$$
by ...
- 18.1k
1
vote
0
answers
47
views
What are all the possible sums (and how often do they occur) of a k-subsequence of an n-sequence of integers?
Let $A_n = \{a_1,\dots,a_n\}$ be a sequence of non-decreasing non-negative integers. Let $P(A_n,k)$ be the set of all subsequences of $A_n$ of length $k$. Given $n,k\in\mathbb Z_{\geqslant0}$ with $n\...
- 1,272
0
votes
0
answers
32
views
How to prove this identity about Sylvestered partitions of n into m parts such that ...
Let us say a partition $\lambda$ is $Sylvestered$ if the smallest part to appear an even number of times ($0$ is an even number) is even. For each set of positive integers $T$, define $S(T\,;m,\,n)$ ...
- 21
3
votes
1
answer
140
views
Prove: Exactly a quarter of 3-part partitions of numbers >2 equal to 0, 2, 10 mod 12 will make a triangle.
Consider perimeters $>2$ equal to $0$, $2$, or $10 \mod(12)$. The sequence starts $10, 12, 14, 22, 24, 26, 34, 36, 38, 46, 48, ...$ and we can look at the three part partitions that make triangles. ...
- 21.4k
4
votes
2
answers
154
views
A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?
Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why,
$$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$
$$\sum_{n=0}^\infty P(n) q_4^{n+1}\approx\...
- 54.9k
0
votes
1
answer
119
views
General term of this sequence
I wanted to know the General term or the function to generate this sequence I found on OEIS.
It is the number of ways to express $2n+1$ as $p+2q$; where $p$ and $q$ can be odd prime number and even ...
- 827
8
votes
3
answers
693
views
How to prove it? (one of the Rogers-Ramanujan identities)
Prove the following identity (one of the Rogers-Ramanujan identities) on formal power series by interpreting each side as a generating function for partitions:
$$1+\sum_{k\geq1}\frac{z^k}{(1-z)(1-z^2)...
- 825
1
vote
1
answer
322
views
Number of perturbations of the Jordan form
I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation.
For example, if a Jordan form consists of a single cell $2 \times ...
- 41