Named after Johann Faulhaber, the rows of Faulhaber's triangle are the coefficients of polynomials that represent sums of integer powers, which are extracted from Faulhaber's formula:








∑

k
=
1


n



k

p


=


1

p
+
1




∑

j
=
0


p





(



p
+
1

j


)




B

j



n

p
+
1
−
j




{\displaystyle \sum _{k=1}^{n}k^{p}={1 \over p+1}\sum _{j=0}^{p}{p+1 \choose j}B_{j}n^{p+1-j}}




where 




B

n




{\displaystyle B_{n}}

 is the nth-Bernoulli number.



The first 5 rows of Faulhaber's triangle, are:

    1
  1/2  1/2
  1/6  1/2  1/3
    0  1/4  1/2  1/4
-1/30    0  1/3  1/2  1/5



Using the third row of the triangle, we have:






∑

k
=
1


n



k

2


=


1
6


n
+


1
2



n

2


+


1
3



n

3




{\displaystyle \sum _{k=1}^{n}k^{2}={1 \over 6}n+{1 \over 2}n^{2}+{1 \over 3}n^{3}}





 Task
 show the first 10 rows of Faulhaber's triangle.
 using the 18th row of Faulhaber's triangle, compute the sum: 




∑

k
=
1


1000



k

17




{\displaystyle \sum _{k=1}^{1000}k^{17}}

 (extra credit).
 See also

 Bernoulli numbers
 Evaluate binomial coefficients
 Faulhaber's formula (Wikipedia)
 Faulhaber's triangle (PDF)


#lang racket
(require math/number-theory)
 
(define (second-bernoulli-number n)
  (if (= n 1) 1/2 (bernoulli-number n)))
 
(define (faulhaber-row:formulaic p)
  (let ((p+1 (+ p 1)))
    (reverse
     (for/list ((j (in-range p+1)))
       (* (/ p+1) (second-bernoulli-number j) (binomial p+1 j))))))
 
(define (sum-k^p:formulaic p n)
  (for/sum ((f (faulhaber-row:formulaic p)) (i (in-naturals 1)))
    (* f (expt n i))))
 
(module+ main
  (map faulhaber-row:formulaic (range 10))
  (sum-k^p:formulaic 17 1000))
 
(module+ test
  (require rackunit)
  (check-equal? (sum-k^p:formulaic 17 1000)
                (for/sum ((k (in-range 1 (add1 1000)))) (expt k 17))))