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Rosetta Code Faulhaber Example

[The following task description was taken from rosettacode.org:]

In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers.

# http://rosettacode.org/wiki/Faulhaber's_formula
faulhaber_example is

  bernoulli (n i32) num.fraction i32
    pre
      n >= 0
  =>
    local_mutate : mutate is
    local_mutate.instate_self (()->
      a := local_mutate.env.new (array n+1 (i -> 1 ⁄ 1))
      for
        m in 0..n
      do
        a <- (a.get.put m (1  ⁄  (m+1)))
        for k in m..1 : -1 do
          a <- (a.get.put (k-1) ((a.get[k - 1] - a.get[k]) * (k ⁄ 1)))
      if n != 1
        a.get[0]
      else
        -a.get[0])


  binomial(n, k i32)
    pre
      n > 0
      k >= 0
    => factorial n / (factorial k * factorial n-k)


  factorial(n i32)
    =>
    for
      res := 1, res * i
      i in 1..n
    do
    else
      res


  faulhaber(p i32) =>
    yak "$p : "
    q := 1  ⁄  (p+1)
    for
      j in 0..p
      sign := 1⁄1, sign * (-1⁄1)
    do
      b := binomial(p+1, j)  ⁄  1
      coeff := q * sign / (1⁄1) * b * bernoulli j
      if coeff != (0⁄1)
        if j = 0
          if coeff = (1⁄1)
          else
            if coeff = (-1⁄1)
              yak "-"
            else
              yak "$coeff"
        else
            if coeff = (1⁄1)
              yak " + "
            else
              if coeff = (-1⁄1)
                yak " - "
              else
                if coeff > (0⁄1)
                  yak " + $coeff"
                else
                  yak " - {-coeff}"
        pwr := p + 1 - j
        if pwr > 1
          yak "n^$pwr"
        else
          yak "n"

  for i in 0..11 do
    faulhaber i
    say ""

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# http://rosettacode.org/wiki/Faulhaber's_formula
faulhaber_example is
  bernoulli (n i32) num.fraction i32
    pre
      n >= 0
  =>
    local_mutate : mutate is
    local_mutate.instate_self (()->
      a := local_mutate.env.new (array n+1 (i -> 1 ⁄ 1))
      for
        m in 0..n
      do
        a <- (a.get.put m (1  ⁄  (m+1)))
        for k in m..1 : -1 do
          a <- (a.get.put (k-1) ((a.get[k - 1] - a.get[k]) * (k ⁄ 1)))
      if n != 1
        a.get[0]
      else
        -a.get[0])
  binomial(n, k i32)
    pre
      n > 0
      k >= 0
    => factorial n / (factorial k * factorial n-k)
  factorial(n i32)
    =>
    for
      res := 1, res * i
      i in 1..n
    do
    else
      res
  faulhaber(p i32) =>
    yak "$p : "
    q := 1  ⁄  (p+1)
    for
      j in 0..p
      sign := 1⁄1, sign * (-1⁄1)
    do
      b := binomial(p+1, j)  ⁄  1
      coeff := q * sign / (1⁄1) * b * bernoulli j
      if coeff != (0⁄1)
        if j = 0
          if coeff = (1⁄1)
          else
            if coeff = (-1⁄1)
              yak "-"
            else
              yak "$coeff"
        else
            if coeff = (1⁄1)
              yak " + "
            else
              if coeff = (-1⁄1)
                yak " - "
              else
                if coeff > (0⁄1)
                  yak " + $coeff"
                else
                  yak " - {-coeff}"
        pwr := p + 1 - j
        if pwr > 1
          yak "n^$pwr"
        else
          yak "n"
  for i in 0..11 do
    faulhaber i
    say ""
 
 
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last changed: 2025-05-12