scholarly journals Binomial Coefficients and Triangular Numbers

2021 ◽  
Vol 2 (3) ◽  
pp. 22-24
Author(s):  
Kantaphon Kuhapatanakul ◽  
Antony G. Shannon
Keyword(s):  
Binomial Coefficients ◽  
Triangular Numbers ◽  
Alternating Sums

We produce formulas of sums the product of the binomial coefficients and triangular numbers. And we apply our formula to prove an identity of Wang and Zhang. Further, we provide an analogue of our identity for the alternating sums.

scholarly journals Congruences involving alternating sums related to harmonic numbers and binomial coefficients

2020 ◽  
Vol 26 (4) ◽  
pp. 39-51
Author(s):  
Laid Elkhiri ◽  
◽  
Miloud Mihoubi ◽  
Abdellah Derbal ◽  
◽  
...  
Keyword(s):  
Prime Number ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Arithmetic Properties ◽  
Alternating Sums ◽  
Generalized Harmonic Numbers

In 2017, Bing He investigated arithmetic properties to obtain various basic congruences modulo a prime for several alternating sums involving harmonic numbers and binomial coefficients. In this paper we study how we can obtain more congruences modulo a power of a prime number p (super congruences) in the ring of p-integer \mathbb{Z}_{p} involving binomial coefficients and generalized harmonic numbers.

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.

Tiling Proofs of Some Formulas for the Pell Numbers of Odd Index

Integers ◽  
2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Mark Shattuck
Keyword(s):  
Diophantine Equation ◽  
Binomial Coefficients ◽  
Combinatorial Interpretation ◽  
Pell Numbers ◽  
Odd Index ◽  
Alternating Sums

AbstractWe provide tiling proofs of several algebraic formulas for the Pell numbers of odd index, all of which involve alternating sums of binomial coefficients, as well as consider polynomial generalizations of these formulas. In addition, we provide a combinatorial interpretation for a Diophantine equation satisfied by the Pell numbers of odd index.

Tribinomial coefficients and Catalan numbers

2009 ◽  
Vol 93 (528) ◽  
pp. 449-455 ◽  
Author(s):  
Thomas Koshy ◽  
Mohammad Salmassi
Keyword(s):  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Triangular Numbers ◽  
Interesting Family ◽  
Positive Integers

The concept of the ordinary binomial coefficientcan be employed to construct an interesting family of positive integers. Such a family was introduced around 1974 by W. Hansell using the triangular numbers where we call them tribinomial coefficients since they are binomial coefficients for triangular numbers. To this end, first we define corresponding to and For example,

scholarly journals Sums of products of generalized Fibonacci and Lucas numbers

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
Zvonko Čerin
Keyword(s):  
Recent Work ◽  
Recurrence Relation ◽  
Fixed Number ◽  
Binomial Coefficients ◽  
Natural Numbers ◽  
Lucas Numbers ◽  
Alternating Sums ◽  
Explicit Formulae ◽  
Fibonacci And Lucas Numbers ◽  
Sums Of Products

AbstractIn this paper we obtain explicit formulae for sums of products of a fixed number of consecutive generalized Fibonacci and Lucas numbers. These formulae are related to the recent work of Belbachir and Bencherif. We eliminate all restrictions about the initial values and the form of the recurrence relation. In fact, we consider six different groups of three sums that include alternating sums and sums in which terms are multiplied by binomial coefficients and by natural numbers. The proofs are direct and use the formula for the sum of the geometric series.

Sign in / Sign up

Export Citation Format

Share Document