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Padovan numbers as sums over partitions into odd parts
- Cristina Ballantine^{1}Email author and
- Mircea Merca^{2}
https://doi.org/10.1186/s13660-015-0952-5
© Ballantine and Merca 2015
Received: 30 September 2015
Accepted: 21 December 2015
Published: 4 January 2016
Abstract
Recently it was shown that the Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. In this paper, we introduce a similar representation for the Padovan numbers. As a corollary, we derive an infinite family of double inequalities.
Keywords
MSC
1 Introduction
A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. The terms of the sequence are called the parts of the partition.
Theorem 1
Proof
By Theorem 1 it is clear that the number of integer partitions of n into odd parts greater than 1 is less than or equal to the nth Padovan number, for any positive integer n. As seen in [8, 9], this inequality can be improved by exploring the number of multinomial coefficients over integer partitions into odd parts greater than 1. As a consequence of these facts, a family of double inequalities will be introduced in the paper.
We note that the inequalities proved in this article are, in fact, analytic statements. However, to our knowledge, there are no analytic proofs for these types of results.
2 On the multinomial coefficients over the partitions into odd parts greater than 1
We use the convention that \(q'(0)=1\) and \(q'(n)=0\) for any negative integer n.
Similar to [9], Theorem 2, we obtain the following.
Theorem 2
Next, we relate \(Q_{k}(n)\) and \(Q'_{k}(n)\) in order to derive the generating function for \(Q'_{k}(n)\) if \(k=1\) or a prime power.
Proposition 1
Proof
By the same argument, \(Q_{p^{r}}(n)\) equals to the number of positive integer solutions \((x,y)\) of (7) with x, y odd and \(x\neq y\). Moreover, \(Q'_{p^{r}}(n)\) equals to the number of positive integer solutions \((x,y)\) of (7) with x, y odd, \(x,y>1\), and \(x\neq y\).
Note that if \(Q_{p^{r}}(n)>0\), then \((1,y)\) with y odd is a solution to (7). In addition, \((x,1)\) with x odd is a solution to (7) if and only if \(p^{r}-1\) divides \(n-1\). Thus, (6) follows. □
Theorem 3
3 A family of double inequalities
Theorem 4
Proof
Theorem 5
Proof
For \(0< x<1\), we remark that the sequence \(\{A_{k}(x)\}_{k>0}\) is strictly monotonically increasing. In addition, Theorems 4 and 5 can be written as follows.
Theorem 6
The following inequality follows directly from Theorem 6.
Corollary 1
Another family of upper bounds for \(\sum_{n=1}^{\infty }{x^{3n}}/{(1-x^{2n})}\) can easily be derived by (5) if we use Theorem 3.
Theorem 7
Proof
4 Other inequalities
Theorem 8
Proof
As in [8], Theorem 7, we have the following.
Theorem 9
Proof
One can similarly consider the cases \(k=3, 4,5\) in Theorem 2 to obtain stronger but more complicated inequalities involving the generating function for the number of partitions into odd parts greater than 1 and the generating function for the number of odd divisors greater than 1.
5 Results and discussion
Corollary 2
Proof
It is still an open problem to give a formula for \(Q'_{k}(n)\) when k is not a prime power.
6 Concluding remarks
We remark that this infinite sum can be replaced by a series which converges much faster for \(|x|<1\).
Theorem 10
Proof
Finally, we note that when the problem of finding a closed form for the generating function of \(Q'_{k}(n)\) for arbitrary k will be solved, then further, stronger inequality families will follow by the methods used in this article.
Declarations
Acknowledgements
The second author wishes to thank the College of the Holy Cross for its hospitality during the writing of this article. The authors thank the anonymous referees for the careful reading of the manuscript and for their useful comments. This work was partially supported by a grant from the Simons Foundation (#245997).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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