The figurate number triangle is Pascal's triangle written in a square grid and padded with zeros, as written by Jakob Bernoulli (Smith 1984). The figurate number triangle therefore has entries
(1)
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where is the row number, the column number, and a binomial coefficient. Written out explicitly (beginning each row with ),
(2)
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Then we have the sum identities
(3)
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(4)
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(5)
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(6)
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