Linear Summing Formulas of Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers

On generalized averaged Gaussian formulas

We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions w(x) ≡ w(α,β)(x) = (1− x)α(1 + x)β (α, β > −1) we give a necessary and sufficient ...

Some summation formulas involving harmonic numbers and generalized harmonic numbers

On generalized fuzzy numbers

This paper first improves Chen and Hsieh’s definition of generalized fuzzy numbers, which makes it the generalization of definition of fuzzy numbers. Secondly, in terms of the generalized fuzzy numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the λ-cutting sets or generalized λ-cutting sets, so that the generalized fuzzy numbers are integrat...

Generalized Catalan Numbers: Linear Recursion and Divisibility

We prove a linear recursion for the generalized Catalan numbers Ca(n) := 1 (a−1)n+1 ( an n ) when a ≥ 2. As a consequence, we show p |Cp(n) if and only if n 6= p−1 p−1 for all integers k ≥ 0. This is a generalization of the well-known result that the usual Catalan number C2(n) is odd if and only if n is a Mersenne number 2 k − 1. Using certain beautiful results of Kummer and Legendre, we give a...

Several Explicit and Recursive Formulas for the Generalized Motzkin Numbers

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.