Using “Pascal’s” triangle to sum kth
powers of consecutive integers
Al-Bahir fi'l Hisab (Shining Treatise on Calculation), alSamaw'al, Iraq, 1144
Siyuan Yujian (Jade Mirror of the Four Unknowns), Zhu
Shijie, China, 1303
Maasei Hoshev (The Art of the Calculator), Levi ben
Gerson, France, 1321
Ganita Kaumudi (Treatise on Calculation), Narayana
Pandita, India, 1356
1 x 0  1
1 x 
1
 1 x
1 x 2  1  2x  x 2
1 x 
 1 3x  3x  x
1 x 
 1 4x  6x 2  4x 3  x 4
1 x 
 1 5x  10x  10 x  5x  x
3
4
5
2
2
3
3
4
5
1 x 0  1
1 x 
1
 1 x
1 x 2  1  2x  x 2
1 x 
 1 3x  3x  x
1 x 
 1 4x  6x 2  4x 3  x 4
1 x 
 1 5x  10x  10 x  5x  x
3
4
5
2
2
3
3
4
5
1 x 0  1
1 x 
1
 1 x
1 x 2  1  2x  x 2
1 x 
 1 3x  3x  x
1 x 
 1 4x  6x 2  4x 3  x 4
1 x 
 1 5x  10x  10 x  5x  x
3
4
5
2
2
k  k 1 k  2
  
 

k   k   k 
3
3
4
5
n n  1
    

k  k  1
1  2 
   
k  k 
0  0 
k  1 k
 
  
 k  k

0
n  n  1
    

k  k  1
1  2 
   
k  k 
0  0 
k  1 k
 
  
 k  k

n  n  1
    

k  k  1
0
Note that the binomial coefficient j
choose k is a polynomial in j of degree k.
1
j 
   Pk  j   j  j 1 j  2
k 
k!
 j  k  1
All the coefficients are
positive integers.
Can we find a simple way
of generating them?
Can we discover what they
count?
HP(k,i ) is the House-Painting number
1
2
3
5
6
7
4
8
It is the number of ways of painting k houses using
exactly i colors.
j 
 j 
j  HP(k,k)  HP(k,k  1)

k
k 1
k
 j 
 HP(k,k  2)

k  2 
j
 HP(k,1) 
1
j k is the number of ways of painting
k houses when we have j colors to choose
from at each house, and we don' t care
whether or not all the colors are used.
1
2
1
6
6
HP(k,k)  k!
1
24 36 14 1
HP(k,1)  1
HP(k,i ) is the House-Painting number
1
2
3
5
6
7
4
8
HP(k,i)  i HPk 1,i   HPk 1,i 1
1
HP(k,k)  k!
2
1
6
6
1
24
36
14
+
X4
HP(k,1)  1
1
120 240 150 30 1
1
HP(k,k)  k!
2
1
6
6
1
24
36
14
+
HP(k,1)  1
X3
1
120 240 150 30 1
1
HP(k,k)  k!
2
1
6
6
1
24
36
14
HP(k,1)  1
+
1
X2
120 240 150 30 1
1
1
2
1
6
6
1
24
36
14
1
120 240 150 30 1
1
1
1
3
1
1
6
7
1
10
25
1
15 1
HP(k,i) is always divisible by i!
(number of ways of permuting the colors)
HP(k,i) / i! = S(k,i) = Stirling number of the second kind
1  2 
   
k  k 
0  0 
k  1 k
 
  
 k  k

0
n  n  1
    

k  k  1
1
 n  1nn  1n  2 n  3
5
3
 n  1nn  1n  2 
2
7
 n  1n n 1
3
1
 n  1n
2