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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
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-
-
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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