Faulhaber’s formula expresses the sum of the th power of the first integers as a function of and .
The are known as the triangular numbers, and the are the square pyramidal numbers, for obvious reason.
During the last 30 years, I have had to establish it by hand for moderate on several occasions, and found the task tedious and error prone every time. Maybe some of you who tried experienced this too. I recently discovered an easy approach that I wanted to share.
As can be seen, the expression is a polynomial of degree . It must be so, because the first order finite difference is a polynomial of degree (namely ). And it has no independent term, given that . Also observe that the coefficient of the leading term is (just like the primitive of ), and the sum of the coefficients is (by ).
The coefficients are closely related to the Bernouilli numbers. A detailed description is given in Wikipedia.
The sum of the cubes
For the sake of the explanation, we shall consider the case of , not too simple, nor too complex. As said, the sum of the cubes gives rise to a fourth degree polynomial of four terms:
The trick that makes it easy is to take the difference of and , which we know to be . We will expand the binomials using the binomial coefficients.
After simplification, we can identify the coefficients of the powers of , and we form this system:
(You should recognize Pascal’s triangle with alternating signs and a diagonal missing.)
Divide every equation by the coefficient of its rightmost term:
This triangular system is readily solved row by row, and as announced:
Using the code
I have coded in Python the computation of the Faulhaber polynomial for arbitrary . The coefficients being fractions, I handle them as such (pairs of integers), keeping the fractions simple by means of the gcd algorithm (Euclid).
def Simplify(a, b):
# Divide a and b by their gcd
c, d= a, b
while d != 0:
c, d= d, c % d
return (a / c, b / c)
To carry out all computation with fractions, it is enough to implement a SAXPY operation (multiply and add):
def SAXPY(A, X, Y):
# Compute A.X + Y, where A, X and Y are fractions
S= Simplify(A * X, A * X)
S= Simplify(S * Y + S * Y, S * Y)
For ease of programming, I precomputed Pascal’s triangle up to the desired level, in a bidimensional array.
# Maximum order of the formula
# Setup Pascal triangle
for i in range(1, K + 2):
for j in range(1, i):
# Recurrence relation
Pascal[i].append(Pascal[i-1][j-1] + Pascal[i-1][j])
The core of the computation, solving the triangula system, is straigthforward:
# Compute the Faulhaber polynomial
for k in range(K + 1):
# Initialize the leading coefficient
S= [(1, k+1)]
# Compute the next coefficients from the triangular system
for i in range(k):
T= (0, 1)
for j in range(i + 1):
# Accumulate, with alternating signs
T= SAXPY(S[j], Simplify(Pascal[k + 1 - j][k - 1 - i], (i - k if (i + j) & 1 else k - i)), T)
Here we are:
S0(n) = n
S1(n) = n^2/2 + n/2
S2(n) = n^3/3 + n^2/2 + n/6
S3(n) = n^4/4 + n^3/2 + n^2/4
S4(n) = n^5/5 + n^4/2 + n^3/3 - n/30
S5(n) = n^6/6 + n^5/2 + 5n^4/12 - n^2/12
S6(n) = n^7/7 + n^6/2 + n^5/2 - n^3/6 + n/42
S7(n) = n^8/8 + n^7/2 + 7n^6/12 - 7n^4/24 + n^2/12
S8(n) = n^9/9 + n^8/2 + 2n^7/3 - 7n^5/15 + 2n^3/9 - n/30
S9(n) = n^10/10 + n^9/2 + 3n^8/4 - 7n^6/10 + n^4/2 - 3n^2/20
S10(n) = n^11/11 + n^10/2 + 5n^9/6 - n^7 + n^5 - n^3/2 + 5n/66
S11(n) = n^12/12 + n^11/2 + 11n^10/12 - 11n^8/8 + 11n^6/6 - 11n^4/8 + 5n^2/12
S12(n) = n^13/13 + n^12/2 + n^11 - 11n^9/6 + 22n^7/7 - 33n^5/10 + 5n^3/3 - 691n/2730
S13(n) = n^14/14 + n^13/2 + 13n^12/12 - 143n^10/60 + 143n^8/28 - 143n^6/20 + 65n^4/12 - 691n^2/420
Points of Interest
It looks so easy now !
This is the first version.