All Questions
9
questions
1
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0
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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,
$$\...
- 51
1
vote
1
answer
143
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Generating function of ordered odd partitions of $n$.
Let the number of ordered partitions of $n$ with odd parts be $f(n)$. Find the generating function $f(n)$ .
My try : For $n=1$ we have $f(1)=1$, for $n=2$, $f(2)=1$, for $n=3$, $f(3)=2$, for $n=4$, $...
user1137804
7
votes
1
answer
167
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How to prove the following resummation identity for Erdős–Borwein constant?
Question:
How to prove
$$\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
1
vote
2
answers
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
5
votes
3
answers
1k
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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 $...
0
votes
1
answer
28
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How does one handle series generating functions with multiple equals signs?
How would somebody walk through this equation? I'm looking for $q(n)$. If I'm given an input of 10, for example, how would this play out? The two equals signs is throwing me off. Does the result of ...
2
votes
0
answers
192
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Is there a generating function for this sequence?
The sequence is:
1, 2, 3, 5, 8, 12, 18, 25, 35, 50, 69, 93, 126, 167, 220, 290, 377, 486, 627, 800, 1017, 1290, 1623, 2032, 2542, 3161, 3917, 4843,...
It is related to partitions of $n$. It is a ...
- 351
0
votes
0
answers
181
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A generating function $G(x)=-\frac{\frac{1}{x^5}(1+\frac{1}{x})(1-\frac{1}{x^2})}{((1-\frac{1}{x})(1-\frac{1}{x^3}))^2}$ related to partitions of $6n$
Fix a sequence $a_n={n+2\choose 2}$ of triangular numbers with the initial condition $a_0=1$,such that
$1,3,6,10,15,21,\dots$
given by
$F(x)=\frac{1}{(1-x)^3}=\sum_{n=0}^{\infty} a_n x^n\tag1$
...
- 2,813
8
votes
3
answers
693
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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)...
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