Trinomial Triangle

The trinomial Triangle (English, about Trinominales triangle) is a variation on Pascal's triangle. The difference is that an entry is the sum of the three (rather like in real Pascal's triangle of the two) is printed above the entries. So far has been able to enforce no generally accepted German term for the rather low mathematical relevance. In the German edition of the book " Chess and Mathematics" by Yevgeny Gik it is called " Pascal's 3- arithmetic triangle".

For the - th entry in the th row has the name

Established. The rows are counted starting with the entries in the -th row with starting up. Thus, the middle node has index, and the symmetry is determined by the formula

Expressed.

Properties

The -th row corresponds to the coefficients of the polynomial expansion of the - th power, ie a special trinomial:

Or symmetrically

Hence also the designation Trinomialkoeffizienten and the relationship with the multinomial:

Furthermore, contain interesting consequences in the diagonals, such as the triangular numbers.

The sum of the elements of the -th row is.

Recursion

The Trinomialkoeffizienten can be calculated using the following recursion formula:

Is being put on and.

The middle entries

The sequence of the middle entries ( sequence A002426 in OEIS )

Has been investigated by Euler: it is explicitly given by

The corresponding generating function is

Euler also noted the exemplum memorabile inductionis fallacis ( notable example of deceptive induction):

With the Fibonacci sequence. For larger, however, the relationship is wrong. George Andrews explained this by the general identity

Chess Mathematics

The triangle corresponds to the number of possible paths of a chess king, he can take ( with a minimum number of moves ) to reach from the top cell in the grid that with the same number.

Importance in combinatorics

The coefficient of the polynomial expansion of indicates how many different ways there are to randomly select cards from a pack of two identical card games each different card. If you have for example two card games with the cards A, B, C, so it looks like this:

In particular, this means for the number of different hands in a double header.

Alternatively, the number of this possibility also calculated by totaling the number of couples in the hand; for there are ways and for the remaining cards there are ways so that this results in the following relationship to the binomial coefficients:

For example, applies

In the example above this then corresponds to the selection of two cards with the 3 options 0 pair (AB, AC, BC ) and the 3 ways with a pair (AA, BB, CC).