Trinomial expansion
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by
where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n.[1] The trinomial coefficients are given by
This formula is a special case of the multinomial formula for m = 3. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.[2]
Properties
The number of terms of an expanded trinomial is the triangular number
where n is the exponent to which the trinomial is raised.[3]
Example
An example of a trinomial expansion with is :
gollark: +>addtag apioforms-class-υ.
gollark: ```sqliteWITH RECURSIVE xaxis(x) AS (VALUES(-2.0) UNION ALL SELECT x+0.05 FROM xaxis WHERE x<1.2), yaxis(y) AS (VALUES(-1.0) UNION ALL SELECT y+0.1 FROM yaxis WHERE y<1.0), m(iter, cx, cy, x, y) AS ( SELECT 0, x, y, 0.0, 0.0 FROM xaxis, yaxis UNION ALL SELECT iter+1, cx, cy, x*x-y*y + cx, 2.0*x*y + cy FROM m WHERE (x*x + y*y) < 4.0 AND iter<28 ), m2(iter, cx, cy) AS ( SELECT max(iter), cx, cy FROM m GROUP BY cx, cy ), a(t) AS ( SELECT group_concat( substr(' .+*#', 1+min(iter/7,4), 1), '') FROM m2 GROUP BY cy )SELECT '`' || '`' || '`' || group_concat(rtrim(t),x'0a') || '`' || '`' || '`' FROM a;```
gollark: ++supported_langs sql
gollark: Maybe make epicbot respond to virtual messages in virtual channels?
gollark: They do not in #apionet. However, they're considered to be from a bot.
See also
- Binomial expansion
- Pascal's pyramid
- Multinomial coefficient
- Trinomial triangle
References
- Koshy, Thomas (2004), Discrete Mathematics with Applications, Academic Press, p. 889, ISBN 9780080477343.
- Harris, John; Hirst, Jeffry L.; Mossinghoff, Michael (2009), Combinatorics and Graph Theory, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 146, ISBN 9780387797113.
- Rosenthal, E. R. (1961), "A Pascal pyramid for trinomial coefficients", The Mathematics Teacher, 54 (5): 336–338.
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