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(12C+) Bernoulli Number
07-28-2019, 12:02 AM (This post was last modified: 08-25-2019 02:59 PM by Albert Chan.)
Post: #5
RE: (12C+) Bernoulli Number
I lookup "A Source Book in Mathematics", chapter "On the Bernoulli numbers":

B(n) is just sum of k^n formula linear term coefficient.

Example, this is how B(6) is calculated, by doing k^6 forward difference
(see thread: https://www.hpmuseum.org/forum/thread-12...#pid110972)

1 64 729 4096 15625 46656 117649 // value of 1^6 to 7^6
63 665 3367 11529 31031 70993     // forward differences
602 2702 8162 19502 39962
2100 5460 11340 20460
3360 5880 9120
2520 3240
720

Sum of k^6 formula = \(1\binom{n}{1}+63\binom{n}{2}+602\binom{n}{3}+2100\binom{n}{4}+3360\binom{n}{5}​+2520\binom{n}{6}+720\binom{n}{7}\)

B(6) = Linear term coefficient = 1/1 - 63/2 + 602/3 - 2100/4 + 3360/5 - 2520/6 + 720/7 = 1/42

Correction:
B(n) is sum of k^n formula, Σ(i^k, i = 0 to n-1) linear term coefficient
B(1) = linear coefficient of n(n+1)/2 - n = 1/2 - 1 = -1/2
(see http://www.mikeraugh.org/Talks/Bernoulli...n-LACC.pdf, slide 31 to 34)

The Python code (next post), work for n=1 because it flipped sign for all odd n.
The "bug" actually fix the sign Smile
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Messages In This Thread
(12C+) Bernoulli Number - Gamo - 07-27-2019, 06:41 AM
RE: (12C+) Bernoulli Number - Albert Chan - 07-27-2019, 12:41 PM
RE: (12C+) Bernoulli Number - Gamo - 07-27-2019, 01:40 PM
RE: (12C+) Bernoulli Number - John Keith - 07-27-2019, 07:49 PM
RE: (12C+) Bernoulli Number - Albert Chan - 07-28-2019 12:02 AM
RE: (12C+) Bernoulli Number - John Keith - 07-28-2019, 11:21 AM
RE: (12C+) Bernoulli Number - Albert Chan - 07-28-2019, 01:08 AM
RE: (12C+) Bernoulli Number - Gamo - 07-28-2019, 02:29 AM
RE: (12C+) Bernoulli Number - Albert Chan - 07-31-2019, 05:14 PM



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