The Fibonacci numbers ,
obey the following recurrence relations,
and so on. As a number triangle, the coefficients are,
See Ron Knott’s article on the fibonomials, so-called since the above is reminiscent of the binomial triangle. However, I found another set of recurrence relations can be given as,
etc. As a number triangle,
Compare the two triangles. Notice how, for odd powers, the same coefficients appear, though moved up by one odd power. I have no explanation for the phenomenon, other than the fact that I’ve seen several instances already of a “recycled” polynomial appearing in many contexts.