Funny Factorials and Slick Sums
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08-05-2019, 03:06 PM
(This post was last modified: 08-07-2019 12:39 PM by Albert Chan.)
Post: #2
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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 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 → s6(n) = (6 n^7 - 21 n^6 + 21 n^5 - 7 n^3 + n) / 42 see threads: Bernoulli Numbers, Sum of Powers |
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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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