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Figurate Number Triangle

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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

a_(ij)=(i; j),
(1)

where i is the row number, j the column number, and (i; j) a binomial coefficient. Written out explicitly (beginning each row with j=0),

[1 0 0 0 0 0 0 ...; 1 1 0 0 0 0 0 ...; 1 2 1 0 0 0 0 ...; 1 3 3 1 0 0 0 ...; 1 4 6 4 1 0 0 ...; 1 5 10 10 5 1 0 ...; 1 6 15 20 15 6 1 ...; 1 7 21 35 35 21 7 ...; | | | | | | | ...]
(2)

Then we have the sum identities

sum_(j=0)^(i)a_(ij)=2^i
(3)
sum_(j=1)^(i)a_(ij)=2^i-1
(4)
sum_(i=0)^(n)a_(ij)=a_((n+1),(j+1))
(5)
=(n+1)/(j+1)a_(nj).
(6)

See also

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

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References

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

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Figurate Number Triangle

Cite this as:

Weisstein, Eric W. "Figurate Number Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FigurateNumberTriangle.html

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