How to resolve the algorithm Faulhaber's formula step by step in the Factor programming language

Published on 12 May 2024 09:40 PM
#Factor

How to resolve the algorithm Faulhaber's formula step by step in the Factor programming language

Table of Contents

Problem Statement

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.

Generate the first 10 closed-form expressions, starting with p = 0.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Faulhaber's formula step by step in the Factor programming language

Source code in the factor programming language

USING: formatting kernel math math.combinatorics math.extras
math.functions regexp sequences ;

: faulhaber ( p -- seq )
    1 + dup recip swap dup <iota>
    [ [ nCk ] [ -1 swap ^ ] [ bernoulli ] tri * * * ] 2with map ;

: (poly>str) ( seq -- str )
    reverse [ 1 + "%un^%d" sprintf ] map-index reverse " + " join ;

: clean-up ( str -- str' )
    R/ n\^1\z/ "n" re-replace            ! Change n^1 to n.
    R/ 1n/ "n" re-replace                ! Change 1n to n.
    R/ \+ -/ "- " re-replace             ! Change + - to - .
    R/ [+-] 0n(\^\d+ )?/ "" re-replace ; ! Remove terms of zero.

: poly>str ( seq -- str ) (poly>str) clean-up ;

10 [ dup faulhaber poly>str "%d: %s\n" printf ] each-integer

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