Algebra

Figurate Numbers

The 𝑛^th term of any order is the sum of 𝑛 terms of the preceding order. The 𝑛^th term of the 𝑟^th order is 𝑛(𝑛+1)⋯(𝑛+𝑟−2)(𝑟−1)!=𝐻(𝑛,𝑟−1) By (98) 289 The sum of 𝑛 terms is 𝑛(𝑛+1)⋯(𝑛+𝑟−1)𝑟!=𝐻(𝑛,𝑟) OrderFigurate Numbers𝑛th term
11,1,1,1,1,1,1 2123456𝑛 3136101521𝑛(𝑛+1)1⋅2 41410203556𝑛(𝑛+1)(𝑛+2)1⋅2⋅3 515153570126𝑛(𝑛+1)(𝑛+2)(𝑛+3)1⋅2⋅3⋅4 6162156126252𝑛(𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)1⋅2⋅3⋅4⋅5 290

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive