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Tagged with integer-partitions sequences-and-series
45
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Find generating series on set of descending sequences, with weight function as taking sum of sequence
Given the set of all sequences of length k with descending (not strictly, so $3,3,2,1,0$ is allowed) terms of natural numbers (including $0$), $S_k$, and the weight function $w(x)$ as taking the sum ...
- 183
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1
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54
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Formula for number of monotonically decreasing sequences of non-negative integers of given length and sum?
What is a formula for number of monotonically decreasing sequences of non-negative integers of given length and sum? For instance, if length k=3 and sum n=5, then these are the 5 sequences that meet ...
- 15
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60
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Generating Function for Modified Multinomial Coefficients
The multinomial coefficients can be used to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + ...} \right)^n}$ in the basis of monomial symmetric polynomials (MSP). For example,
$$\...
- 51
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Prove convergence of partition sequence [duplicate]
We are given a partition of a positive integer $x$. Each step, we make a new partition of $x$ by decreasing each term in the partition by 1, removing all 0 terms, and adding a new term equal to the ...
5
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1
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238
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Prove that the general formula for a sequence $a_n$ is $\frac{(-1)^n}{n!}$
Here is a sequence $a_n$ where the first five $a_n$ are:
$a_1=-\frac{1}{1!}$
$a_2=-\frac{1}{2!}+\frac{1}{1!\times1!}$
$a_3=-\frac{1}{3!}+\frac{2}{2!\times1!}-\frac{1}{1!\times1!\times1!}$
$a_4=-\frac{...
1
vote
1
answer
57
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What is the 11th unordered combination of natural numbers that add upto 6 in the partition function?
So, I was making unordered combinations of natural numbers which add upto a certain natural number. I was able to go till 6 when I got to know about the partition function. I was pleased to see that ...
- 177
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126
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An identity related to the series $\sum_{n\geq 0}p(5n+4)x^n$ in Ramanujan's lost notebook
While browsing through Ramanujan's original manuscript titled "The Lost Notebook" (the link is a PDF file with 379 scanned pages, so instead of a click it is preferable to download) I found ...
- 88.3k
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1
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142
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Generating function of ordered odd partitions of $n$.
Let the number of ordered partitions of $n$ with odd parts be $f(n)$. Find the generating function $f(n)$ .
My try : For $n=1$ we have $f(1)=1$, for $n=2$, $f(2)=1$, for $n=3$, $f(3)=2$, for $n=4$, $...
user1137804
2
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2
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81
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Show that series converges by estimating number of partitions into distinct parts
I need some help with solving the following problem: Let $Q(n)$ be the number of partitions of $n$ into distinct parts. Show that $$\sum_{n=1}^\infty\frac{Q(n)}{2^n}$$ is convergent by estimating $Q(n)...
- 155
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167
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How to prove the following resummation identity for Erdős–Borwein constant?
Question:
How to prove
$$\sum_{m=1}^{\infty}\left(1-\prod_{j=m}^{\infty}(1-q^j)\right) = \sum_{n=1}^{\infty}\frac{q^n}{1-q^n} \tag{1}$$
for all $q \in \mathbb{C}$ such that $\left|q\right| < 1$?
...
- 3,132
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34
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How might I go about (dis)proving a conjecture involving RMS, integer compositions, and contiguous subsequences of finite sequences?
I'm here for some help to prove or disprove a (possibly trivial) conjecture concerning compositions (i.e. ordered partitions) of the natural number $n$, and the contiguous subsequences that they ...
1
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2
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193
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What is the closed form solution to the sum of inverse products of parts in all compositions of n?
My question is exactly as in the title:
What is the generating function or closed form solution to the sum of inverse products of parts in all compositions of $n$?
This question was inspired by just ...
- 171
2
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1
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38
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Sum of $k-tuple$ partition of $d$. $k$ and $d$ are both positive integers.
I am trying to understand this paper by Bondy and Jackson and in it I found the following calculation in which I cannot figure out how we got from 2nd last step to last step. I have referred the paper ...
5
votes
1
answer
371
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What's the sequence $3,9,24,21,36,30,75,120,270,462,837,1320,2085,\ldots$?
In order to find a formula of the partition of an integer into 5 parts (see $(6)$ below), I find the sequence
$$S_1: 3,9,24,21,36,30,75,120,270,462,837,1320,2085,\ldots \tag1$$
It's clear that $S_1$ ...
5
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3
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Proving that odd partitions and distinct partitions are equal
I am working through The Theory of Partitions by George Andrews (I have the first paperback edition, published in 1998).
Corollary 1.2 is a standard result that shows that the number of partitions of $...
