k^rの和の公式
(%i1) nusum(k^1,k,1,n);
n (n + 1)
(%o1)
---------
2
(%i2) nusum(k^2,k,1,n);
n (n + 1) (2 n + 1)
(%o2)
-------------------
6
(%i3) nusum(k^3,k,1,n);
2 2
n (n + 1)
(%o3)
-----------
4
(%i4) nusum(k^4,k,1,n);
2
n (n + 1) (2 n + 1) (3 n + 3 n - 1)
(%o4)
------------------------------------
30
(%i5) nusum(k^5,k,1,n);
2 2 2
n (n + 1) (2 n + 2 n - 1)
(%o5)
----------------------------
12
(%i6) nusum(k^6,k,1,n);
4 3
n (n + 1) (2 n + 1) (3 n + 6 n - 3 n + 1)
(%o6)
-------------------------------------------
42
(%i7) nusum(k^7,k,1,n);
2 2
4 3 2
n (n + 1) (3 n + 6 n - n - 4 n + 2)
(%o7)
----------------------------------------
24
(%i8) nusum(k^8,k,1,n);
6 5
4 3 2
n (n + 1) (2 n + 1) (5 n +
15 n + 5 n - 15 n - n + 9 n - 3)
(%o8)
----------------------------------------------------------------
90
(%i9) nusum(k^9,k,1,n);
2 2
2
4 3 2
n (n + 1) (n + n - 1) (2 n + 4 n -
n - 3 n + 3)
(%o9)
-----------------------------------------------------
20
(%i10) nusum(k^10,k,1,n);
2
(%o10) (n (n + 1) (2 n + 1) (n + n - 1)
6 5
4 3 2
(3 n + 9 n + 2 n - 11 n + 3 n + 10 n -
5))/66
(%i11) nusum(k^11,k,1,n);
(%o11)
2 2
8 7
6 5
4 3 2
n (n + 1) (2 n + 8 n +
4 n - 16 n - 5 n + 26 n - 3 n - 20 n + 10)
--------------------------------------------------------------------------
24
(%i12) nusum(k^12,k,1,n);
10
9
8
7 6
(%o12) (n (n + 1) (2 n + 1) (105 n + 525 n + 525
n - 1050 n - 1190 n
5
4
3 2
+ 2310 n + 1420 n - 3285 n - 287 n + 2073 n -
691))/2730
(%i13) nusum(k^13,k,1,n);
2
2
10
9
8
7
6 5
(%o13) (n (n + 1) (30 n + 150 n + 125
n - 400 n - 326 n + 1052 n
4
3 2
+ 367 n - 1786 n + 202 n + 1382 n - 691))/420
(%i14) nusum(k^14,k,1,n);
12
11
10
9
8 7
(%o14) (n (n + 1) (2 n + 1) (3 n + 18 n + 24
n - 45 n - 81 n + 144 n
6
5
4
3 2
+ 182 n - 345 n - 217 n + 498 n + 44 n -
315 n + 105))/90
(%i15) nusum(k^15,k,1,n);
2 2
12
11
10
9
8
7 6
(%o15) (n (n + 1) (3 n + 18 n + 21
n - 60 n - 83 n + 226 n + 203 n
5
4
3 2
- 632 n - 226 n + 1084 n - 122 n - 840 n +
420))/48
(%i16) nusum(k^16,k,1,n);
14
13
12
11 10
(%o16) (n (n + 1) (2 n + 1) (15 n + 105 n + 175
n - 315 n - 805 n
9
8
7
6
5
4 3
+ 1365 n + 2775 n - 4845 n - 6275 n +
11835 n + 7485 n - 17145 n
2
- 1519 n + 10851 n - 3617))/510
(%i17) nusum(k^17,k,1,n);
2
2 14
13
12
11
10 9
(%o17) (n (n + 1) (10 n + 70 n +
105 n - 280 n - 565 n + 1410 n
8
7
6
5
4
3 2
+ 2165 n - 5740 n - 5271 n + 16282 n +
5857 n - 27996 n + 3147 n
+ 21702 n - 10851))/180
(%i18) nusum(k^18,k,1,n);
16
15
14
13 12
(%o18) (n (n + 1) (2 n + 1) (105 n + 840 n +
1680 n - 2940 n - 9996 n
11
10
9
8
7 6
+ 16464 n + 48132 n - 80430 n -
167958 n + 292152 n + 380576 n
5
4
3 2
- 716940 n - 454036 n + 1039524 n + 92162
n - 658005 n + 219335))/3990
(%i19) nusum(k^19,k,1,n);
2
2
16
15
14
13 12
(%o19) (n (n + 1) (42 n + 336 n +
616 n - 1568 n - 4263 n
11
10
9
8
7 6
+ 10094 n + 22835 n - 55764 n -
87665 n + 231094 n + 213337 n
5
4
3 2
- 657768 n - 236959 n + 1131686 n - 127173
n - 877340 n + 438670))/840
(%i20) nusum(k^20,k,1,n);
18
17
16 15
(%o20) (n (n + 1) (2 n + 1) (165 n + 1485 n +
3465 n - 5940 n
14
13
12
11
10 9
- 25740 n + 41580 n + 163680
n - 266310 n - 801570 n + 1335510 n
8
7
6
5 4
+ 2806470 n - 4877460 n - 6362660 n + 11982720
n + 7591150 n
3 2
- 17378085 n - 1540967 n + 11000493 n - 3666831))/6930
(%i21)