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[Triangle of] Fibonacci polynomials



















1 x 4 + 10 x 3 + 15 x 2 + 7 x + 1 


5 x 4 + 20 x 3 + 21 x 2 + 8 x + 1 


1 x 5 + 15 x 4 + 35 x 3 + 28 x 2 + 9 x + 1 


6 x 5 + 35 x 4 + 56 x 3 + 36 x 2 + 10 x + 1 

The sequence of
Fibonacci polynomials ^{[2]} is a
sequence of polynomials defined by the
recurrence relation
 ${\begin{array}{l}\displaystyle {F_{n}(x):={\begin{cases}0,&{\mbox{if }}n=0,\\1,&{\mbox{if }}n=1,\\F_{n1}(x)+x\,F_{n2}(x),&{\mbox{if }}n\geq 2,\end{cases}}}\end{array}}$
where
If you look at the
Fibonacci polynomials triangle, you will see that the rising diagonals corresponding to odd
are the "
(1, 1)Pascal polynomials". And the column of degree
have
natural numbers as coefficients, the column of degree
have
triangular numbers as coefficients, the column of degree
have
tetrahedral numbers as coefficients, and so on... (Cf. rows of
(1, 1)Pascal triangle, i.e.
Pascal triangle.)
This yields the infinite sequence of finite sequences

{{0}, {1}, {1}, {1, 1}, {2, 1}, {1, 3, 1}, {3, 4, 1}, {1, 6, 5, 1}, {4, 10, 6, 1}, {1, 10, 15, 7, 1}, {5, 20, 21, 8, 1}, {1, 15, 35, 28, 9, 1}, {6, 35, 56, 36, 10, 1}, ... }, 
whose concatenation yields the infinite sequence (see A102426)

{ 0, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 4, 1, 1, 6, 5, 1, 4, 10, 6, 1, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, ... }. 
Fibonacci rabbits per generation
If we interpret the Fibonacci numbers
as the number of
[malefemale] pairs of Fibonacci rabbits at the beginning of the
th month, where each pair that is at least 2 months old begets a
[malefemale] pair of newborn Fibonacci rabbits, then the coefficient
of the term
of the Fibonacci polynomial
indicates the number of
[malefemale] pairs of Fibonacci rabbits which belong to the
th generation.
Tree of newborn (red) and mature (green) "[malefemale] pairs of Fibonacci rabbits" (node labels show the generation )
0
0
0
1
0
1
1
0
1
1
1
2
0
1
1
1
2
1
2
2



For example
 $F_{6}(x)=3x^{2}+4x+1\,$
means that at the beginning of the 6^{ th} month, we have
 3 pairs belonging to the 2^{ nd} generation (offsprings of the offsprings of the original pair),
 4 pairs belonging to the 1^{st} generation (offsprings of the original pair),
 1 pair belonging to the 0^{ th} generation (the original pair),
for a total of 8 [malefemale] pairs of Fibonacci rabbits.
At the beginning of the
th month, we have
generations, i.e. generations
to
, of
[malefemale] pairs of Fibonacci rabbits. The original pair, i.e. the 0
^{ th} generation, was a
[malefemale] pair of newborn Fibonacci rabbits, i.e. 0 months old, which was dropped on the island at the beginning of the first month. (At the beginning of the 0
^{ th} month, there were no rabbits.)
Reading the rows of generation labels (the labels of row
are the labels of row
appended with the incremented labels of row
) lefttoright yields the infinite sequence (see A??????)

{0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 2, 2, ...}. 
Reading the rows of generation labels righttoleft yields the infinite sequence (see
A200650 )

{0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 2, 1, 1, 1, 0, ...}. 
Formulae
 ${\begin{array}{l}\displaystyle {F_{n}(x)=\sum _{k=0}^{\lceil {\frac {n}{2}}\rceil 1}T_{(1,1)}\left({{\bigg \lfloor }{\frac {n}{2}}{\bigg \rfloor }+k},{{\bigg \lfloor }{\frac {n1}{2}}{\bigg \rfloor }k}\right)\,x^{2k+1(n{\bmod {2}})}=\sum _{k=0}^{\lceil {\frac {n}{2}}\rceil 1}{\binom {\lfloor {\frac {n}{2}}\rfloor +k}{\lfloor {\frac {n1}{2}}\rfloor k}}\,x^{2k+1(n{\bmod {2}})},\quad n\geq 1,}\end{array}}$
where
T(1, 1) (n, k) = ( ^{n}_{k} ) 
is the term from column
of row
of the
(1, 1)Pascal triangle (i.e.
Pascal triangle), and
is a
binomial coefficient.
Generating function
The ordinary generating function for the sequence of Fibonacci polynomials is
 ${\begin{array}{l}\displaystyle {G_{\{F_{n}(x)\}}(t)=\sum _{n=0}^{\infty }F_{n}(x)\,t^{n}={\frac {t}{1t\,(x+t)}}={\frac {t}{1(t^{1}x^{1}+t^{2}x^{0})}}.}\end{array}}$
We may also observe the following relation
 ${\begin{array}{l}\displaystyle {{\frac {t}{1t\,(x+t)}}=\sum _{n=0}^{\infty }{\left({\frac {t}{1t^{2}}}\right)}^{n+1}x^{n}=\sum _{n=0}^{\infty }{\left(\sum _{k=0}^{\infty }t^{2k+1}\right)}^{n+1}x^{n},}\end{array}}$
where
 ${\begin{array}{l}\displaystyle {\sum _{k=0}^{\infty }t^{2k+1}={\frac {t}{1t^{2}}}.}\end{array}}$
See also
 A049310 Triangle of coefficients of Chebyshev polynomials (exponents in increasing order). Unsigned triangle has Fibonacci polynomials (definition according to MathWorld) as row polynomials.
Notes
 ↑ Weisstein, Eric W., Fibonacci Polynomial, from MathWorld—A Wolfram Web Resource.
 ↑ MathWorld has a different definition^{[1]} for the sequence of Fibonacci polynomials (which are related to the Chebyshev polynomials, see A049310), i.e.
 ${\begin{array}{l}\displaystyle {F_{n}(x):={\begin{cases}1,&{\mbox{if }}n=1,\\x,&{\mbox{if }}n=2,\\x\,F_{n1}(x)+F_{n2}(x),&{\mbox{if }}n\geq 3.\end{cases}}}\end{array}}$