Complementary Riordan arrays. (English) Zbl 1288.05015
Summary: Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas for the elements of many recursive matrices.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 11B83 | Special sequences and polynomials |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 15B99 | Special matrices |
Online Encyclopedia of Integer Sequences:
Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).Triangle T(n,k), read by rows, defined by T(n,k) = binomial(2*n,n-k).
Array read by rows: right-hand side of triangle A027907 of trinomial coefficients.
Inverse of the Motzkin triangle A064189.
Riordan array (1, x*(1+x)/(1-x^3)).
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