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Figurate Number Triangle

A Pascal's Triangle written in a square grid and padded with zeroes, as written by Jakob Bernoulli (Smith 1984). The figurate number triangle therefore has entries

a_{ij}={i\choose j},

where $i$ is the row number, $j$ the column number, and ${i\choose j}$ a Binomial Coefficient. Written out explicitly (beginning each row with $j=0$),

1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots\cr
1 & ...
...vdots & \vdots & \vdots & \vdots & \vdots & \ddots\cr}}\right]

Then we have the sum identities
$\displaystyle \sum_{j=0}^i a_{ij}$ $\textstyle =$ $\displaystyle 2^i$  
$\displaystyle \sum_{j=1}^i a_{ij}$ $\textstyle =$ $\displaystyle 2^i-1$  
$\displaystyle \sum_{i=0}^n a_{ij}$ $\textstyle =$ $\displaystyle a_{(n+1),(j+1)}={n+1\over j+1} a_{nj}.$  

See also Binomial Coefficient, Figurate Number, Pascal's Triangle


Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 86, 1984.

© 1996-9 Eric W. Weisstein