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

Published on 12 May 2024 09:40 PM
#Echolisp

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

Source code in the echolisp programming language

(lib 'math) ;; for bernoulli numbers
(string-delimiter "")

;; returns list of polynomial coefficients
(define (Faulhaber p)
	(cons 0
	(for/list ([k (in-range p -1 -1)])
		(* (Cnp (1+ p) k) (bernoulli k)))))

;; prints formal polynomial
(define (task (pmax 10))
    (for ((p pmax)) 
    (writeln p '→  (/ 1 (1+ p)) '* (poly->string 'n (Faulhaber p)))))
    
;; extra credit - compute sums
(define (Faulcomp n p)
	(printf "Σ(1..%d) n^%d = %d" n p (/  (poly n (Faulhaber p)) (1+ p) )))

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