## Odd powers of Fibonacci numbers

The Fibonacci numbers $F_n$, $F_n = 0, 1, 1, 2, 3, 5, 8, 13, \dots$

obey the following recurrence relations, \begin{aligned}&F_n-F_{n-1}-F_{n-2} = 0\\[1.5mm]&F_n^2-2F_{n-1}^2-2F_{n-2}^2+F_{n-3}^2 = 0\\[1.5mm]&F_n^3-3F_{n-1}^3-6F_{n-2}^3+3F_{n-3}^3+F_{n-4}^3 = 0\\[1.5mm]&F_n^4-5F_{n-1}^4-15F_{n-2}^4+15F_{n-3}^4+5F_{n-4}^4-F_{n-5}^4 = 0\\[1.5mm]&F_n^5-8F_{n-1}^5-40F_{n-2}^5+60F_{n-3}^5+40F_{n-4}^5-8F_{n-5}^5-F_{n-6}^5 = 0\end{aligned}

and so on.  As a number triangle, the coefficients are, \begin{aligned} &1;\; \bold{1, -1,-1}=0\\ &2;\; 1, -2, -2, \;1=0\\ &3;\; \bold{1, -3, -6, \;3, \;1}=0\\ &4;\; 1, -5, -15, 15, \;5, \;-1=0\\ &5;\; \bold{1, -8, -40, 60, 40, -8, -1}=0 \end{aligned}

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, \begin{aligned} &F_{n-1}^2-F_{n+1}^2 = -F_{2n}\\[1.5mm] &F_{n-1}^3-F_{n}^3-F_{n+1}^3 = -F_{3n}\\[1.5mm] &F_{n-2}^4+3F_{n-1}^4-3F_{n+1}^4-F_{n+2}^4 = -6F_{4n}\\[1.5mm] &F_{n-2}^5-3F_{n-1}^5-6F_{n}^5+3F_{n+1}^5+F_{n+2}^5 = 6F_{5n}\\[1.5mm] &F_{n-3}^6+4F_{n-2}^6-20F_{n-1}^6+20F_{n+1}^6-4F_{n+2}^6-F_{n+3}^6 = -120F_{6n}\\[1.5mm] &F_{n-3}^7-8F_{n-2}^7-40F_{n-1}^7+60F_{n}^7+40F_{n+1}^7-8F_{n+2}^7-F_{n+3}^7 = -240F_{7n}\\[1.5mm]\end{aligned}

etc.  As a number triangle, \begin{aligned} &2;\; 1, -1 =-1\\ &3;\; \bold{1, -1,-1}=-1\\ &4;\; 1, \;\;\;3, -3, \;-1=-6\\ &5;\; \bold{1, -3, -6, \;3, \;\;1}\;=\;6\\ &6;\; 1, \;\;\;4, -20, 20, -4, -1=-120\\ &7;\; \bold{1, -8, -40, 60, 40, -8, -1}=-240 \end{aligned}

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.