Something for sci.math's amateur mathematicians?
- From: "Dave L. Renfro" <renfr1dl@xxxxxxxxx>
- Date: 20 Aug 2006 14:19:42 -0700
Occasionally a poster asks if there is anything an amateur
mathematician could work on that has not already been
thoroughly explored. I don't know if the following qualifies,
but I thought I'd throw it out in case someone is interested.
Let T_n = 1 + 2 + 3 + ... + n = n(n+1)/2. Then T_n is the
n'th triangular number. The n'th tetrahedral number is
T_1 + T_2 + T_3 + ... + T_n = n(n+1)(n+2)/6 [1]. We can
repeat this process to form the sum of the first n tetrahedral
numbers to get the n'th 4-tetrahedral number n(n+1)(n+2)(n+3)/24,
and so on for the higher order versions [2].
[1] http://mathworld.wolfram.com/TetrahedralNumber.html
http://en.wikipedia.org/wiki/Tetrahedral_number
http://mathforum.org/workshops/usi/pascal/pascal_tetrahedral.html
[2] http://www.math.toronto.edu/mathnet/questionCorner/tetnumbers.html
Apparently, this is all fairly well known. However, I wonder if
the analogous situation for multiplication (or exponentiation)
replacing addition has been studied.
For example, let's call 1*2*3*...*n = n! the 1'st order factorial
of n, 1!*2!*3!*...*n! the 2'nd order factorial of n, and so on.
Are there any interesting mathematical issues going on with these
higher order factorial numbers?
Dave L. Renfro
.
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