Trinomial coefficients are a generalization of binomial coefficients to three variables.
Another use of trinomial coefficients is the coefficients of x^k in the expansion of (1 + x + 1/x)^n.
Trinomial coefficients can be visualized in a triangle analogous to Pascal’s triangle, where each entry is the sum of the three entries above it.
Trinomial coefficients relate to the moves of a king on a chessboard, representing the number of paths a king can take to a square using the minimal number of moves.