The trinomial triangle is a variation of
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
. 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
-th entry of the
-th row is denoted by
:
.
Rows are counted starting from 0. The entries of the
-th row are indexed starting with
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
:
Properties
The
-th row corresponds to the coefficients in the
polynomial expansion
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions tha ...
of the expansion of the
trinomial
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.
Examples of trinomial expressions
# 3x + 5y + 8z with x, y, z variables
# 3t + 9s^2 + 3y^3 with t, s, y variables
# 3ts + 9t + 5s with t, s variables
# ...
raised to the
-th power:
:
or, symmetrically,
:
,
hence the alternative name trinomial coefficients because of their relationship to the
multinomial coefficients:
:
Furthermore, the diagonals have interesting properties, such as their relationship to the
triangular numbers
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
.
The sum of the elements of
-th row is
.
Recurrence formula
The trinomial coefficients can be generated using the following
recurrence formula:
:
,
:
for
,
where
for
and
.
Central trinomial coefficients
The middle entries of the trinomial triangle
: 1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, …
were studied by
Euler and are known as central trinomial coefficients.
The
-th central trinomial coefficient is given by
:
Their
generating function is
:
Euler noted the following ''exemplum memorabile inductionis fallacis'' ("notable example of fallacious induction"):
:
for
,
where
is the ''n''-th
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
. For larger
, however, this relationship is incorrect.
George Andrews explained this fallacy using the general identity
[George Andrews, Three Aspects for Partitions. ''Séminaire Lotharingien de Combinatoire'', B25f (1990]
Online copy
/ref>
:
Applications
In chess
The triangle corresponds to the number of possible paths that can be taken by the king
King is the title given to a male monarch in a variety of contexts. The female equivalent is queen, which title is also given to the consort of a king.
*In the context of prehistory, antiquity and contemporary indigenous peoples, the tit ...
in a game of chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
. The entry in a cell represents the number of different paths (using a minimum number of moves) the king can take to reach the cell.
In combinatorics
The coefficient of in the expansion of gives the number of different ways to draw cards from two identical sets of playing cards each.[Andreas Stiller: ''Pärchenmathematik. Trinomiale und Doppelkopf.'' ("Pair mathematics. Trinomials and the game of ''Doppelkopf''"). ]c't
''c't'' – ' (''Magazine for Computer Technology'') is a German computer magazine, published by the Heinz Heise publishing house.
The 5.71 meter high tower from the 587 published c't editions up to the 30th anniversary has been in the foyer ...
Issue 10/2005, p. 181ff For example, from two sets of the three cards A, B, C, the different drawings are:
For example,
:.
In particular, this provides the formula for the number of different hands in the card game ''Doppelkopf
Doppelkopf (, lit. ''double-head''), sometimes abbreviated to Doko, is a trick-taking card game for four players. The origins of this game are not well known; it is only recorded from the early 20th century and it is assumed that it originated f ...
''.
Alternatively, it is also possible to arrive at this expression by considering the number of ways of choosing pairs of identical cards from the two sets, which is the binomial coefficient . The remaining cards can then be chosen in ways, which can be written in terms of the binomial coefficients as
:.
The example above corresponds to the three ways of selecting two cards without pairs of identical cards (AB, AC, BC) and the three ways of selecting a pair of identical cards (AA, BB, CC).
References
Further reading
* {{cite journal, author=Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, title=Observationes analyticae ("Analytical observations") , journal=Novi Commentarii Academiae Scientiarum Petropolitanae , volume=11 , year=1767 , pages=124–143 , url=http://www.math.dartmouth.edu/~euler/pages/E326.html
Discrete mathematics
Triangles of numbers
Factorial and binomial topics