Funny Factorials and Slick Sums
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08-07-2019, 01:57 PM
(This post was last modified: 08-08-2019 12:41 PM by Albert Chan.)
Post: #3
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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 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) |
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Messages In This Thread |
Funny Factorials and Slick Sums - Albert Chan - 08-05-2019, 12:21 AM
RE: Funny Factorials and Slick Sums - Albert Chan - 08-05-2019, 03:06 PM
RE: Funny Factorials and Slick Sums - Albert Chan - 08-07-2019 01:57 PM
RE: Funny Factorials and Slick Sums - pier4r - 08-07-2019, 04:45 PM
RE: Funny Factorials and Slick Sums - Albert Chan - 11-02-2021, 02:59 PM
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