Results 111 to 120 of about 174,940 (323)

Factors of alternating convolution of the Gessel numbers [PDF]

Notes on Number Theory and Discrete Mathematics
The Gessel number P(n,r) is the number of lattice paths in the plane with (1,0) and (0,1) steps from (0,0) to (n+r, n+r-1) that never touch any of the points from the set {(x,x)∈ℤ²:x≥r}. We show that there is a close relationship between Gessel numbers P(
Jovan Mikić
doaj   +1 more source

On sums of binomial coefficients and their applications

Discrete Mathematics, 2008
In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For example, we show that if $n\ge m-1$ then $$\sum_{i=0}^{\lfloor(m-1)/2\rfloor}(-1)^i\binom{m-1-i}i [n-2i,r-i]_m=2^{n-
openaire   +3 more sources

Remark on a conjecture of Erdős on binomial coefficients [PDF]

, 1970
Andrzej Mạkowski
openalex   +1 more source

Alu Overexpression Leads to an Increased Double‐stranded RNA Signature in Dermatomyositis

Arthritis &Rheumatology, Accepted Article.
Objective Dermatomyositis is an autoimmune condition characterized by a high interferon signature of unknown etiology. Because coding sequences constitute <1.2% of our genomes, there is a need to explore the role of the non‐coding genome in disease pathogenesis.
Rayan Najjar, Andrew Mammen, Tomas Mustelin   +2 more
wiley   +1 more source

Generating Functions for Binomial Series Involving Harmonic-like Numbers

Mathematics
By employing the coefficient extraction method, a class of binomial series involving harmonic numbers will be reviewed through three hypergeometric F12(y2)-series.
Chunli Li, Wenchang Chu
doaj   +1 more source

Representations and binomial coefficients

Annals of Representation Theory
For a root system R, a field K and a "choice of coefficients in K" we define a category of graded spaces with operators and study some of its properties. Then we assume that the coefficients are given by quantum binomials. We use basic arithmetic properties of binomial coefficients (such as q-versions of Lucas' theorem and the Pfaff-Saalschütz identity)
openaire   +2 more sources

A note on balancing binomial coefficients

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2015
In 2014, T. Komatsu and L. Szalay studied the balancing binomial coefficients. In this paper, we focus on the following Diophantine equation $$\binom{1}{5}+\binom{2}{5}+...+\binom{x-1}{5}=\binom{x+1}{5}+...+\binom{y}{5}$$ where $y>x>5$ are integer unknowns. We prove that the only integral solution is $(x,y)=(14,15)$. Our method is mainly based on
openaire   +5 more sources

ON DIVISIBILITY OF BINOMIAL COEFFICIENTS [PDF]

Journal of the Australian Mathematical Society, 2012
AbstractIn this paper, motivated by Catalan numbers and higher-order Catalan numbers, we study factors of products of at most two binomial coefficients.
openaire   +2 more sources

Longitudinal links of negative student–teacher relationships and positive class climate with traditional bullying and cyberbullying perpetration

British Educational Research Journal, EarlyView.
Abstract The current study aimed to investigate whether negative student–teacher relationships and within‐class perceptions of the class climate at the individual level, and positive class climates at the classroom level in fifth grade, were associated with traditional bullying and cyberbullying perpetration 1 year later, in sixth grade, in a sample of
Robert Thornberg, Björn Sjögren, Jun Sung Hong, Claudio Longobardi   +3 more
wiley   +1 more source

Gauss’ Second Theorem for ${}_{2}F_1(1/2)$-Series and Novel Harmonic Series Identities

Mathematics
Two summation theorems concerning the F12(1/2)-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in ...
Chunli Li, Wenchang Chu
doaj   +1 more source