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Pascal's Tetrahedron

Pascal's tetrahedron is the three-dimensional analog of Pascal's triangle. The entries in layer n are the multinomial coefficients

(n!)/(i!j!k!),

for nonnegative integers i, j, and k satisfying i+j+k=n. Each interior entry is the sum of the three adjacent entries above it, as follows from the recurrence relation

(n!)/(i!j!k!)=((n-1)!)/((i-1)!j!k!)+((n-1)!)/(i!(j-1)!k!)+((n-1)!)/(i!j!(k-1)!),

where a term with a negative index is interpreted as zero. The entries in layer n are the coefficients in the multinomial series for (x+y+z)^n.

See also

Multinomial Coefficient, Multinomial Series, Pascal's Triangle, Tetrahedron

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References

Németh, L. "Tetrahedron Trinomial Coefficient Transform." Integers 19, Article A41, 2019. https://math.colgate.edu/~integers/t41/t41.pdf.

Cite this as:

Weisstein, Eric W. "Pascal's Tetrahedron." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PascalsTetrahedron.html

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