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factorial fun
Posted:
Dec 18, 2002 8:55 PM


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(n1)(n2)...1 by definition. 2! = 2*1 = n(n1) 3! = 3*2*1 = n(n1)(n2) 4! = 4*3*2*1 = n(n1)(n2)(n3) 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
 markdotmath@earthlink.net  EarthLink: The #1 provider of the Real Internet.
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