Journal of Integer Sequences, Vol. 12 (2009), Article 09.7.4

Hadamard Products and Tilings

Jong Hyun Kim
Department of Mathematics
Brandeis University
Waltham, MA 02454-9110


Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula

\begin{displaymath}\frac{1}{1-ax-x^2}\ast \frac{1}{1-bx-x^2}
= \frac{1-x^2}{1-abx-(2+a^2+b^2)x^2
-abx^3+x^4}, \end{displaymath}

where $*$ denotes the Hadamard product. In a similar way, by considering tilings of a $2\times n$ rectangle with $1\times1$ and $1\times 2$ bricks in the top row, and $1\times1$ and $1\times n$ bricks in the bottom row, we find an explicit formula for the Hadamard product

\begin{displaymath}\frac{1}{1-ax-x^2}\ast \frac{x^m}{1-bx-x^n}.\end{displaymath}

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(Concerned with sequences A000045 A007598 A011973 A093614 A114525.)

Received March 3 2009; revised version received October 20 2009. Published in Journal of Integer Sequences, October 21 2009.

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