Funny Factorials and Slick Sums
08-05-2019, 12:21 AM (This post was last modified: 08-19-2020 11:54 AM by Albert Chan.)
Post: #1
 Albert Chan Senior Member Posts: 1,243 Joined: Jul 2018
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

08-05-2019, 03:06 PM (This post was last modified: 08-07-2019 12:39 PM by Albert Chan.)
Post: #2
 Albert Chan Senior Member Posts: 1,243 Joined: Jul 2018
RE: Funny Factorials and Slick Sums
(08-05-2019 12:21 AM)Albert Chan Wrote:  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

We can automate the process of simplifying.

For s6(n), polynomial linear coefficent = limit(s6(n)/n, n=0):

(-1)6/7 + 15 (-1)5/6 + 65 (-1)4/5 + 90 (-1)3/4 + 31 (-1)2/3 + (-1)1/2
= -1*(1/2 - 2*(31/3 - 3*(90/4 - 4*(65/5 - 5*(15/6 - 6*(1/7))))))
= -1*(1/2 - 2*(31/3 - 3*(90/4 - 4*(65/5 - 5*(23/14)))))
= -1*(1/2 - 2*(31/3 - 3*(90/4 - 4*(67/14))))
= -1*(1/2 - 2*(31/3 - 3*(47/14)))
= -1*(1/2 - 2*(11/42))
= -1*(-1/42)
= 1/42

→ s6(n) / n = (1/7) n6 + (23/14) n5 + (67/14) n4 + (47/14) n3 + (11/42) n2 + (-1/42) n1 + 1/42

We divide n repeatedly, collecting remainder terms, until quotient is linear, since a n1 + b n0 = a n + b

Code:
Synthetic Division, falling factorial form to polynomial
6 105 546 945 434  21   0   0  // 42*s6, falling factorial coefficients
-6 0> 6  69 201 141  11  -1   1      // -6*6+105=69, -5*69+546=201 ...
-5 0> 6  39  45   6  -1   0          // -5*6+69=39, -4*39+201=45 ...
-4 0> 6  15   0   6  -7
-3 0> 6  -3   6   0
-2 0> 6 -15  21
-1 0> 6 -21
Note: 0> signalled first column numbers treated as stack, "popped" when used.

→ s6(n) = (6 n^7 - 21 n^6 + 21 n^5 - 7 n^3 + n) / 42

see threads: Bernoulli Numbers, Sum of Powers
08-07-2019, 01:57 PM (This post was last modified: 08-08-2019 12:41 PM by Albert Chan.)
Post: #3
 Albert Chan Senior Member Posts: 1,243 Joined: Jul 2018
RE: Funny Factorials and Slick Sums
We can generalize falling factorial form polynomial and power form polynomial as Newton form polynomial.
For falling factorial form, offsets = 0,1,2,3, ...
For power form, offsets = 0,0,0,0, ...

Below is the synthetic division, that can convert from 1 set of offsets, to another.

Example: simplify this: S4(n) = -17 + $$16\binom{n+3}{1}-15\binom{n+3}{2}+14 \binom{n+3}{3}-12\binom{n+3}{4}+24\binom{n+3}{5}$$

To setup synthetic division, negate the from-offsets, and put to first column, in reverse order.
For above S4(n), negated from-offsets = 3,2,1,0,-1

Next, put to-offsets to second column.
Since S4(-1) = S4(0) = 0, we know S4(n) has factor n*(n+1), thus to-offsets = -1,0,0,0,0

For each row, to-offset numbers is "locked". Negated from-offsets treated like a stack, "popped" when used.

Code:
24  -12   14  -15   16  -17 // S4 in binomial form
6  -15   70 -225  480 -510 // 30*S4 in falling factorial form
-1 -1>   6  -27   97 -225  255    0 // (-1-1)*6-15=-27, (0-1)*-27+70=97, (1-1)*97-225=-225 ...
0  0>   6  -27   70  -85    0      // (0+0)*6-27=-27, (1+0)*-27+97=70, (2+0)*70-225=-85 ...
1  0>   6  -21   28   -1
2  0>   6   -9    1
3  0>   6    9

6n³ + 9n² + n - 1 = (2n+1) (3n² + 3n - 1)

→ S4(n) = (n)(n+1)(2n+1)(3n² + 3n - 1) / 30

source: "Fundamentals of Numerical Anaylsis" by Stephen G. Kellison, published in 1975
Appendix B: Transformation of polynomial forms. (modified to use only +* for synthetic division)
08-07-2019, 04:45 PM
Post: #4
 pier4r Senior Member Posts: 2,067 Joined: Nov 2014
RE: Funny Factorials and Slick Sums
I love this.

Quote:WARNING! In several of the problems, you may be tempted touse an ellipsis (“. . . ”) to shorten arguments or to indicate that apattern continues. DO NOT do this. All proofs must be complete.Ellipsis probably means you need to use induction or recursion

Wikis are great, Contribute :)
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