Trinomial Triangle

The Number Triangle obtained by starting with a row containing a single ``1'' and the next row containing three 1s and then letting subsequent row elements be computed by summing the elements above to the left, directly above, and above to the right:

$1 \quad\ 1 \quad\ 1$

$1 \quad\ 2 \quad\ 3 \quad\ 2 \quad\ 1$

$1 \quad\ 3 \quad\ 6 \quad\ 7 \quad\ 6 \quad\ 3 \quad\ 1$

$1 \quad 4 \quad 10 \quad 16 \quad 19 \quad 16 \quad 10 \quad 4 \quad 1$

(Sloane's A027907). The th row can also be obtained by expanding and taking coefficients:

$\displaystyle (1+x+x^2)^0$	$\textstyle =$	$\displaystyle 1$
$\displaystyle (1+x+x^2)^1$	$\textstyle =$	$\displaystyle 1+x+x^2$
$\displaystyle (1+x+x^2)^2$	$\textstyle =$	$\displaystyle 1+2x+3x^2+2x^3+x^4$

and so on.

See also Pascal's Triangle

References

Sloane, N. J. A. Sequence A027907 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.


$1 \quad\ 1 \quad\ 1$
$1 \quad\ 2 \quad\ 3 \quad\ 2 \quad\ 1$
$1 \quad\ 3 \quad\ 6 \quad\ 7 \quad\ 6 \quad\ 3 \quad\ 1$
$1 \quad 4 \quad 10 \quad 16 \quad 19 \quad 16 \quad 10 \quad 4 \quad 1$