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ベルヌーイ数とお友達になろう #ロマンティック数学ナイト

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

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ベルヌーイ数とお友達になろう #ロマンティック数学ナイト

  1. 1. @tsujimotter
  2. 2. 2
  3. 3. 1 3 7 13 3
  4. 4. 1 3 7 13 21 +2 +4 +6 +8 4
  5. 5. 5 6 30 42 30 66
  6. 6. 6 6 30 42 30 66
  7. 7. 7 6 30 42 30 66 2 4 6 8 10 6 1, 2, 3, 6
  8. 8. 8 6 30 42 30 66 2 4 6 8 10 6 1, 2, 3, 6 +1 2, 3, 4, 7
  9. 9. 1 -1 1 -1 1 -691 9
  10. 10. 10
  11. 11. 1 + 2 + 3 + · · · + 100 = ? 11
  12. 12. 1 2 3 … 98 99 100 + ) 100 99 98 … 3 2 1 101 101 101 … 101 101 101 100 12
  13. 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. 14
  15. 15. 1k + 2k + 3k + · · · + nk = (n + B)k+1 Bk+1 k + 1 Bk Bk Bk 15
  16. 16. 1k + 2k + 3k + · · · + nk = (n + B)k+1 Bk+1 k + 1 Bk Bk Bk 16
  17. 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. 18. 1 + 2 + 3 + · · · + n = n2 B0 + 2nB1 2 1 + 2 + 3 + · · · + n = n2 + n 2 B0 = 1, B1 = 1 2 18
  19. 19. 1 1 2 1 3 1 4 1 5 1 2 “1” 19
  20. 20. 1 1 2 1 3 1 4 1 5 1 2 1 3 “2” 20
  21. 21. 1 1 2 1 3 1 4 1 5 1 2 1 3 “3” 1 4 21
  22. 22. 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 22
  23. 23. 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 1 6 “1” 23
  24. 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. 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. 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. 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. 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. 29. 10 29
  30. 30. http://tsujimotter.info/irregulars/bernoulli.html 30
  31. 31. NEW!! http://tsujimotter.info/irregulars/faulhaber.html 31
  32. 32. 32
  33. 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. 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. 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. 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. 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. 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. 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. 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. 41. 1 -1 1 -1 1 -691 41
  42. 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. • • • 43

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