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Topic: factorial fun
Replies: 2   Last Post: Dec 22, 2002 8:55 PM

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Mark Schwartz

Posts: 48
Registered: 12/6/04
factorial fun
Posted: Dec 18, 2002 8:55 PM
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Hi All,

I got intrigued by the fun with the question about finding the factorial
that generates P, given P. I played some more with what I posted earlier
regarding an algebraic approach. Look at this:
n! = n(n-1)(n-2)...1 by definition.
2! = 2*1 = n(n-1)
3! = 3*2*1 = n(n-1)(n-2)
4! = 4*3*2*1 = n(n-1)(n-2)(n-3)
etc, etc.
In all cases, the last term is 1.

Notice that the polynomial that results is to the nth power. No surprise.
But for the fun of it, I generated the polynomials to n = 6. What got me -
and I hope someone can give me some insights - is the pattern of numbers
that occurs in the coefficients of the polynomials.

for 2, it's 1,-1
for 3, 1, -3, 2
for 4, 1, -6, 11, -6
for 5, 1, -10, 35, -50, 24
for 6, 1, -15, 85, -225, 274, -120

Several things: (1) the sum of the coefficients in all cases = 0 and (2)
the positive value is 1/2 the factorial of N. Anybody understand the
number theory behind this, or can point me to a reference? ... thanks ...
mark


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