Manuals/calci/PASCAL

MATRIX("PASCAL",order)

• is the size of the Pascal matrix.

Description

• This function returns the matrix of any order with the property of Pascal.
• The Pascal matrix is an infinite matrix containing the binomial coefficients as its elements.
• To obtain a pascal matrix there are three ways: as either an upper-triangular matrix(U), a lower-triangular matrix(L), or a symmetric matrix(S).
• Example for these matrices are

• The amazing relationship of these matrices are: .
• And its determinants also 1.i.e.,
• The Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix.
• The elements of the symmetric Pascal matrix are the binomial coefficients, i.e.

, where n=i+j, r=i.

• In other words,

.

• Here MATRIX("pascal") is showing the pascal matrix of order 3.
• So users can change the order of the matrix also.

Examples

• 1.MATRIX("pascal") =1
• 2.MATRIX("pascal",3)
 1 1 1 1 2 3 1 3 6
• 3.MATRIX("pascal",5)
 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70