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 $j=0$),

$$\left[{\matrix{ 1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots\cr 1 & ... ...vdots & \vdots & \vdots & \vdots & \vdots & \ddots\cr}}\right]$$

Then we have the sum identities

$$\sum_{j=0}^i a_{ij} = 2^i$$

$$\sum_{j=1}^i a_{ij} = 2^i-1$$

$$\sum_{i=0}^n a_{ij} = a_{(n+1),(j+1)}={n+1\over j+1} a_{nj}.$$

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

References

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