The trinomial triangle is a variation of Pascal's triangle. The difference between the two is that an entry in the trinomial triangle is the sum of the three (rather than the two in Pascal's triangle) entries above it:
The k {\displaystyle k} -th entry of the n {\displaystyle n} -th row is denoted by
( n k ) 2 {\displaystyle {n \choose k}_{2}} . Rows are counted starting from 0. The entries of the n {\displaystyle n} -th row are indexed starting with − n {\displaystyle -n} from the left, and the middle entry has index 0. The symmetry of the entries of a row about the middle entry is expressed by the relationship
( n k ) 2 = ( n − k ) 2 {\displaystyle {n \choose k}_{2}={n \choose -k}_{2}}
Properties The n {\displaystyle n} -th row corresponds to the coefficients in the polynomial expansion of the trinomial ( 1 + x + x 2 ) {\displaystyle (1+x+x^{2})} raised to the n {\displaystyle n} -th power:
( 1 + x + x 2 ) n = ∑ j = 0 2 n ( n j − n ) 2 x j = ∑ k = − n n ( n k ) 2 x n + k {\displaystyle \left(1+x+x^{2}\right)^{n}=\sum _{j=0}^{2n}{n \choose j-n}_{2}x^{j}=\sum _{k=-n}^{n}{n \choose k}_{2}x^{n+k}}
or, symmetrically,
( 1 + x + 1 / x ) n = ∑ k = − n n ( n k ) 2 x k {\displaystyle \left(1+x+1/x\right)^{n}=\sum _{k=-n}^{n}{n \choose k}_{2}x^{k}} , hence the alternative name trinomial coefficients because of their relationship to the multinomial coefficients:
( n k ) 2 = ∑ 0 ≤ μ , ν ≤ n μ + 2 ν = n + k n ! μ ! ν ! ( n − μ − ν ) ! {\displaystyle {n \choose k}_{2}=\sum _{\textstyle {0\leq \mu ,\nu \leq n \atop \mu +2\nu =n+k}}{\frac {n!}{\mu !\,\nu !\,(n-\mu -\nu )!}}}
Furthermore, the diagonals have interesting properties, such as their relationship to the triangular numbers. The sum of the elements of n {\displaystyle n} -th row is 3 n {\displaystyle 3^{n}} .
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