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Funny Factorials and Slick Sums
08-05-2019, 12:21 AM (This post was last modified: 08-07-2019 12:53 AM by Albert Chan.)
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Funny Factorials and Slick Sums
Spring 2017 ARML Power Contest, Problems and Solutions.

Example, find s6(n) = Σ(x^6, x = 0 to n-1)

Convert x^6 to falling factorial form, where xn = product(x-k, k = 0 .. n-1)

Code:
Synthetic Division, polynomial to falling factorial form:
   1  0  0  0  0  0  0  // x^6
1> 1  1  1  1  1  1     // = (x-1)(x^5 + x^4 + x^3 + x^2 + 1) + 1
2> 1  3  7 15 31
3> 1  6 25 90
4> 1 10 65
5> 1 15
Note: synthetic divide by (x-0) step skipped, since x^6 = (x-0)*x^5 + 0

x^6 = x6 + 15 x5 + 65 x4 + 90 x3 + 31 x2 + x1

From problem 12, Σ(xm, x=0 to n-1) = nm+1 / (m+1)

s6(n) = n7/7 + 15 n6/6 + 65 n5/5 + 90 n4/4 + 31 n3/3 + n2/2

After simplify, s6(n) = n^7/7 - n^6/2 + n^5/2 - n^3/6 + n/42

Edit: more Power Contest Archive
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Funny Factorials and Slick Sums - Albert Chan - 08-05-2019 12:21 AM



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