Editing Fibonacci polynomials (section)

==Combinatorial interpretation==
[[File:pascal_triangle_fibonacci.svg|thumb|upright=1.25|The coefficients of the Fibonacci polynomials can be read off from a left-justified Pascal's triangle following the diagonals (shown in red). The sums of the coefficients are the Fibonacci numbers.]]
If ''F''(''n'',''k'') is the coefficient of ''x<sup>k</sup>'' in ''F<sub>n</sub>''(''x''), namely 
:<math>F_n(x)=\sum_{k=0}^n F(n,k)x^k,\,</math>
then ''F''(''n'',''k'') is the number of ways an ''n''−1 by 1 rectangle can be tiled with 2 by 1 [[domino]]es and 1 by 1 squares so that exactly ''k'' squares are used.<ref name=BQ141/> Equivalently, ''F''(''n'',''k'') is the number of ways of writing ''n''−1 as an [[Composition (number theory)|ordered sum]] involving only 1 and 2, so that 1 is used exactly ''k'' times. For example F(6,3)=4 and 5 can be written in 4 ways, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, as a sum involving only 1 and 2 with 1 used 3 times. By counting the number of times 1 and 2 are both used in such a sum, it is evident that 
<math>F(n, k)=\begin{cases}\displaystyle\binom{\frac12(n+k-1)}{k} &\text{if }n \not\equiv k \pmod 2,\\[12pt]
0 &\text{else}.
\end{cases}</math>

This gives a way of reading the coefficients from [[Pascal's triangle]] as shown on the right.