Funny Factorials and Slick Sums

[Albert Chan]

(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