A note on the \(k\)-Narayana sequence. (English) Zbl 1349.11031
Summary: In the present article, we define the \(k\)-Narayana sequence of integer numbers. We study recurrence relations and some combinatorial properties of these numbers, and of the sum of their first \(n\) terms. These properties are derived from matrix methods. We also study some relations between the \(k\)-Narayana sequence and convolved \(k\)-Narayana sequence, and permanents and determinants of one type of Hessenberg matrix. Finally, we show how these sequences arise from a family of substitutions.
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11B83 | Special sequences and polynomials |
| 05A15 | Exact enumeration problems, generating functions |
Online Encyclopedia of Integer Sequences:
a(1)=0, a(2)=0, a(3)=1, a(n+1) = a(n) - a(n-2).Expansion of 1/(1-3*x-x^3).
Group the natural numbers into groups (1),(2),(3),(4),(5,6),(7,8,9),... so that the n-th group contains N(n) terms, where N(n) is the Narayana’s cows sequence (A000930). Sequence contains the sum of the terms in the n-th group.
