Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev
polynomials equivalent to the formula
where
![$*$](img2.gif)
denotes the Hadamard product. In a similar way, by considering tilings of a
![$2\times n$](img3.gif)
rectangle
with
![$1\times1$](img4.gif)
and
![$1\times 2$](img5.gif)
bricks in the top row, and
![$1\times1$](img4.gif)
and
![$1\times n$](img6.gif)
bricks in the bottom row,
we find an explicit formula for the
Hadamard product