In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

Soil organic matter (SOM) formation involves microbial transformation of plant materials various quality with physico-chemical stabilisation via soil aggregation. Land use and vegetation type can affect the litter chemistry bioavailability carbon (OC), consequently influence processing OC into SOM. We used 13C nuclear magnetic resonance (13C NMR) hot-water extraction to assess changes in chemic...

the main objective of this study is to swing krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. to fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. consequently, the p...

Abstract Background Understanding drivers of animal biodiversity has been a longstanding aim in evolutionary biology. Insects and fishes represent the largest lineages invertebrates vertebrates respectively, consequently many ideas have proposed to explain this diversity. Natural enemy interactions are often important diversification dynamics, key traits that mediate such may therefore an role ...