The Pascal rhombus and Riordan arrays. (English) Zbl 1443.11023
Summary: In the present article, we first obtain Riordan array expressions for the right half of the Pascal rhombus and the left-bounded rhombus. Then, a combinatorial interpretation based on the 2-generalized Motzkin paths is given for these arrays. Moreover, using the \(k\)-generalized Motzkin paths, we introduce the concept of \(k\)-generalized Pascal rhombus and left-bounded rhombus. Finally, explicit formulas for the generic elements and row sums of the \(k\)-generalized Pascal rhombus and left-bounded rhombus are obtained in terms of \(k\)-bonacci numbers.
MSC:
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
Online Encyclopedia of Integer Sequences:
Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n.Pascal’s ”rhombus” (actually a triangle T(n,k), n >= 0, 0<=k<=2n) read by rows: each entry is sum of 3 terms above it in previous row and one term above it two rows back.
Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0)=1, T(n,k)=0 if n < k, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1).
Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,k), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) (0<=k<=n).
Tribonacci left-bounded rhombic triangle.
Irregular triangle read by rows: a Pascal ”rhombus”, third in the sequence after A059317 and A027907.
