scholarly journals When can the sum of $(1/p)$th of the binomial coefficients have closed form?

10.37236/1336 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Marko Petkovšek ◽  
Herbert S. Wilf
Keyword(s):  
Closed Form ◽  
Nonnegative Integer ◽  
Binomial Coefficients

We find all nonnegative integer $r,s,p$ for which the sum $\sum_{k=rn}^{sn}{pn\choose k}$ has closed form.

scholarly journals Harmonic Numbers and Cubed Binomial Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Polygamma Functions ◽  
Form Representation ◽  
The Given

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

scholarly journals Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1599 ◽  
Author(s):  
Henryk A. Witek ◽  
Johanna Langner
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Kekulé Structures ◽  
Structural Resemblance ◽  
Numerical Tests ◽  
Interface Theory ◽  
Complete Set

We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekuléan benzenoids that can be obtained by overlapping two parallelograms: generalized ribbons Rb, parallelograms M, vertically overlapping parallelograms MvM, horizontally overlapping parallelograms MhM, and intersecting parallelograms MxM. All formulas have the form of multiple sums over binomial coefficients. Three of the formulas are given with a proof based on the interface theory of benzenoids, while the remaining two formulas are presented as conjectures verified via extensive numerical tests. Both of the conjectured formulas have the form of a 2×2 determinant bearing close structural resemblance to analogous formulas for the number of Kekulé structures derived from the John-Sachs theory of Kekulé structures.

scholarly journals Trees and Meta-Fibonacci Sequences

10.37236/218 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Abraham Isgur ◽  
David Reiss ◽  
Stephen Tanny
Keyword(s):  
Closed Form ◽  
Empirical Evidence ◽  
Nonnegative Integer ◽  
Asymptotic Estimates ◽  
Combinatorial Interpretation ◽  
Labeled Trees ◽  
Special Cases ◽  
Fibonacci Sequences

For $k>1$ and nonnegative integer parameters $a_p, b_p$, $p = 1..k$, we analyze the solutions to the meta-Fibonacci recursion $C(n)=\sum_{p=1}^k C(n-a_p-C(n-b_p))$, where the parameters $a_p, b_p$, $p = 1..k$ satisfy a specific constraint. For $k=2$ we present compelling empirical evidence that solutions exist only for two particular families of parameters; special cases of the recursions so defined include the Conolly recursion and all of its generalizations that have been studied to date. We show that the solutions for all the recursions defined by the parameters in these families have a natural combinatorial interpretation: they count the number of labels on the leaves of certain infinite labeled trees, where the number of labels on each node in the tree is determined by the parameters. This combinatorial interpretation enables us to determine various new results concerning these sequences, including a closed form, and to derive asymptotic estimates. Our results broadly generalize and unify recent findings of this type relating to certain of these meta-Fibonacci sequences. At the same time they indicate the potential for developing an analogous counting interpretation for many other meta-Fibonacci recursions specified by the same recursion for $C(n)$ with other sets of parameters.

scholarly journals Second order alternating harmonic number sums

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3511-3524 ◽  
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Second Order ◽  
Integral Representations ◽  
Harmonic Number ◽  
Binomial Coefficients ◽  
Harmonic Numbers

We develop new closed form representations of sums of alternating harmonic numbers of order two and reciprocal binomial coefficients. Moreover we develop new integral representations in terms of harmonic numbers of order two.

scholarly journals FACTORS OF SUMS AND ALTERNATING SUMS INVOLVING BINOMIAL COEFFICIENTS AND POWERS OF INTEGERS

2011 ◽  
Vol 07 (07) ◽  
pp. 1959-1976 ◽  
Author(s):  
VICTOR J. W. GUO ◽  
JIANG ZENG
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Binomial Coefficients ◽  
Alternating Sums ◽  
Divisibility Properties ◽  
Positive Integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers n1,…,nm, nm+1 = n1, and any nonnegative integer r, there holds [Formula: see text] and conjecture that for any nonnegative integer r and positive integer s such that r + s is odd, [Formula: see text] where ε = ±1.

scholarly journals Tournament Sequences and Meeussen Sequences

10.37236/1522 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Matthew Cook ◽  
Michael Kleber
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Nonnegative Integer ◽  
Tree Structure ◽  
Efficient Technique ◽  
Counting Problems ◽  
Asymptotic Growth ◽  
Counting Technique ◽  
Positive Integers

A tournament sequence is an increasing sequence of positive integers $(t_1,t_2,\ldots)$ such that $t_1=1$ and $t_{i+1} \leq 2t_i$. A Meeussen sequence is an increasing sequence of positive integers $(m_1,m_2,\ldots)$ such that $m_1=1$, every nonnegative integer is the sum of a subset of the $\{m_i\}$, and each integer $m_i-1$ is the sum of a unique such subset. We show that these two properties are isomorphic. That is, we present a bijection between tournament and Meeussen sequences which respects the natural tree structure on each set. We also present an efficient technique for counting the number of tournament sequences of length $n$, and discuss the asymptotic growth of this number. The counting technique we introduce is suitable for application to other well-behaved counting problems of the same sort where a closed form or generating function cannot be found.

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