Ronald Brady

Posts: 43
Registered: 12/8/04
Nth Order Pascal Triangle
Posted: Aug 8, 1997 7:07 PM
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Nth Order Pascal's Triangle

Several years ago I presented the following to fellow stu-
dents and to Professors:

Pascal's 2nd Order Triangle



1 1
1 2 1
1 3 3 1
1 4 6 4 1
-
-
-

It will be seen that the sum of the entries in the jth row
above is 2^j.

Pascal's 3rd Order Triangle


1 1 1
1 2 3 2 1
1 3 6 7 6 3 1
1 4 10 16 19 16 10 4 1
-
-
-

It will be seen that the sum of the entries in the jth row
is 3^j. Also

each entry in the jth row ( j = or > 2 ) is the sum of the 3
entries in the (j-1)th row,that appear directly above and to
the left of the given entry. For example 7, in the 3rd row,
is the sum of 2,3 and 2 from the 2nd row. Blank spaces above
an entry are assumed to be occupied by zeroes.

It is conjectured that these ideas may be generalized to the nth
order case.

We all know that the entries in the 2nd order triangle represent
the coefficients in binomial expansions. What about the coeffi-
cents in the 3rd and higher order cases ?

Perhaps these should be called Pascal's Nth Order Trapezoids ?



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