faulhaber_s_triangle.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# License: GPLv3
# Date: 07 February 2017
# https://github.com/trizen

# Generation of Faulhaber's triangle.

# Each row of the triangle represents the coefficients
# for the sum 1^p + 2^p + 3^p + ... + n^p.

# For example, the third row is the solution for:
#  1^2 + 2^2 + 3^2 + ... + n^2
#
# which can be written as:
#   1/3 * (1*n^3 + 3/2*n^2 + 1/2*n^1)

# In general, we have:
#       1^p + 2^p + 3^p + ... + n^p
#
# is equal with:
#       1/(p+1) * (F[0] * n^(p+1) + F[1] * n^p + F[2] * n^(p-1) + ... + F[p] * n^1)

# See also:
#   https://en.wikipedia.org/wiki/Faulhaber%27s_formula

# Inspired by John Conway:
#    https://www.youtube.com/watch?v=Uy1B_eGXQ0g

func faulhaber_triangle(p) {
    gather {
        for j in (0 .. p) {
            take(binomial(p+1, j) * bernoulli(j))
        }
    }
}

for p in (0 .. 11) {
    say faulhaber_triangle(p).map{ '%6s' % .as_rat }.join
}

__END__
  1
  1     1
  1   3/2   1/2
  1     2     1     0
  1   5/2   5/3     0  -1/6
  1     3   5/2     0  -1/2     0
  1   7/2   7/2     0  -7/6     0   1/6
  1     4  14/3     0  -7/3     0   2/3     0
  1   9/2     6     0 -21/5     0     2     0 -3/10
  1     5  15/2     0    -7     0     5     0  -3/2     0
  1  11/2  55/6     0   -11     0    11     0 -11/2     0   5/6
  1     6    11     0 -33/2     0    22     0 -33/2     0     5     0