べき乗和の公式を通して、私の好きな「ベルヌーイ数」を紹介します。 2018年8月12日に開催されたロマンティック数学ナイトプライム＠北海道 #2 で発表した資料です。 http://romanticmathnight.org/642 スライド中で紹介したデモはこちら： ベルヌーイ数計算機 http://tsujimotter.info/irregulars/bernoulli.html べき乗和の公式ジェネレータ http://tsujimotter.info/irregulars/faulhaber.html 発表者：tsujimotter http://tsujimotter.info

1. @tsujimotter

2. 2

3. 1 3 7 13
3

4. 1 3 7 13 21
+2 +4 +6 +8
4

5. 5
6 30 42 30 66

6. 6
6 30 42 30 66

7. 7
6 30 42 30 66
2 4 6 8 10
6 1, 2, 3, 6

8. 8
6 30 42 30 66
2 4 6 8 10
6 1, 2, 3, 6
+1 2, 3, 4, 7

9. 1 -1 1 -1 1 -691
9

10. 10

11. 1 + 2 + 3 + · · · + 100 = ?
11

12. 1 2 3 … 98 99 100
+ ) 100 99 98 … 3 2 1
101 101 101 … 101 101 101
100
12

13. 1 + 2 + 3 + · · · + n =
n(n + 1)
2
12
+ 22
+ 32
+ · · · + n2
=
n(n + 1)(2n + 1)
6
13
+ 23
+ 33
+ · · · + n3
=
n2
(n + 1)2
43
2
1
13

14. 14

15. 1k
+ 2k
+ 3k
+ · · · + nk
=
(n + B)k+1
Bk+1
k + 1
Bk Bk
Bk
15

16. 1k
+ 2k
+ 3k
+ · · · + nk
=
(n + B)k+1
Bk+1
k + 1
Bk Bk
Bk
16

17. k = 1
11
+ 21
+ 31
+ · · · + n1
=
(n + B)2
+ B2
2
Bk
7! Bk
=
n2
B0 + 2nB1
2
=
n2
B0
+ 2n1
B1
+ n0
B2
B2
2
∵
17

18. 1 + 2 + 3 + · · · + n =
n2
B0 + 2nB1
2
1 + 2 + 3 + · · · + n =
n2
+ n
2
B0 = 1, B1 =
1
2
18

19. 1
1
2
1
3
1
4
1
5
1
2 “1”
19

20. 1
1
2
1
3
1
4
1
5
1
2
1
3 “2”
20

21. 1
1
2
1
3
1
4
1
5
1
2
1
3 “3”
1
4
21

22. 1
1
2
1
3
1
4
1
5
1
2
1
3
1
4
1
5
22

23. 1
1
2
1
3
1
4
1
5
1
2
1
3
1
4
1
5
1
6 “1”
23

24. 1
1
2
1
3
1
4
1
5
1
2
1
3
1
4
1
5
1
6 “2”
1
6
24

25. 1
1
2
1
3
1
4
1
5
1
2
1
3
1
4
1
5
1
6
1
6
3
20
25

26. 1
1
2
1
3
1
4
1
5
1
2
1
3
1
4
1
5
1
6
1
6
3
20
B1 =
B0 =
B2 =
26

27. 1
1
2
1
3
1
4
1
5
1
2
1
3
1
4
1
5
1
6
1
6
3
20
B1 =
B0 =
B2 =
27

28. B0 = 1
B1 =
1
2
B2 =
1
6
B3 = 0
B4 =
1
30
B6 =
1
42
B8 =
1
30
B10 =
5
66
B5 = 0
B7 = 0
B9 = 0
10
28

29. 10
29

30. http://tsujimotter.info/irregulars/bernoulli.html
30

31. NEW!!
http://tsujimotter.info/irregulars/faulhaber.html
31

32. 32

33. B0 = 1
B1 =
1
2
B2 =
1
6
B3 = 0
B4 =
1
30
B6 =
1
42
B8 =
1
30
B10 =
5
66
B5 = 0
B7 = 0
B9 = 0
B11 = 0
B12 =
691
2730
B14 =
7
6
B16 =
3617
510
B13 = 0
B15 = 0
B17 = 0
33

34. B0 = 1
B1 =
1
2
B2 =
1
6
B3 = 0
B4 =
1
30
B6 =
1
42
B8 =
1
30
B10 =
5
66
B5 = 0
B7 = 0
B9 = 0
B11 = 0
B12 =
691
2730
B14 =
7
6
B16 =
3617
510
B13 = 0
B15 = 0
B17 = 0
0 B1
34

35. B2 =
1
6
B4 =
1
30
B6 =
1
42
B8 =
1
30
B10 =
5
66
B12 =
691
2730
B14 =
7
6
B16 =
3617
510
35

36. B2 =
1
6
B4 =
1
30
B6 =
1
42
B8 =
1
30
B10 =
5
66
B12 =
691
2730
B14 =
7
6
B16 =
3617
510
36

37. B2 =
1
6
B4 =
1
30
B6 =
1
42
B8 =
1
30
B10 =
5
66
B12 =
691
2730
B14 =
7
6
B16 =
3617
510
37

38. B2 =
1
6
B4 =
1
30
B6 =
1
42
B8 =
1
30
B10 =
5
66
B12 =
691
2730
B14 =
7
6
B16 =
3617
510
38

39. B2
2
=
1
12
B4
4
=
1
120
B10
10
=
1
132
B8
8
=
1
240
B16
16
=
3617
8160
B14
14
=
1
12
B12
12
=
691
32760
B6
6
=
1
252
39

40. B2
2
=
1
12
B4
4
=
1
120
B10
10
=
1
132
B8
8
=
1
240
B16
16
=
3617
8160
B14
14
=
1
12
B12
12
=
691
32760
B6
6
=
1
252
40

41. 1 -1 1 -1 1 -691
41

42. 1 -1 1 -1 1 -691
42
B2
2
=
1
12
B4
4
=
1
120
B8
8
=
1
240
B12
12
=
691
32760
B6
6
=
1
252
B10
10
=
1
132

43. •
•
•
43