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Faulhaber's triangle: Difference between revisions
Added a Scheme implementation.
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-1/30 0 2/9 0 -7/15 0 2/3 1/2 1/9
0 -3/20 0 1/2 0 -7/10 0 3/4 1/2 1/10 </pre>
=={{header|Scheme}}==
{{works with|Chez Scheme}}
<lang scheme>; Return the first row-count rows of Faulhaber's Triangle as a vector of vectors.
(define faulhabers-triangle
(lambda (row-count)
; Calculate and store the value of the first column of a row.
; The value is one minus the sum of all the rest of the columns.
(define calc-store-first!
(lambda (row)
(vector-set! row 0
(do ((col-inx 1 (1+ col-inx))
(col-sum 0 (+ col-sum (vector-ref row col-inx))))
((>= col-inx (vector-length row)) (- 1 col-sum))))))
; Generate the Faulhaber's Triangle one row at a time.
; The element at row i >= 0, column j >= 1 (both 0-based) is the product
; of the element at i - 1, j - 1 and the fraction ( i / ( j + 1 ) ).
; The element at column 0 is one minus the sum of all the rest of the columns.
(let ((tri (make-vector row-count)))
(do ((row-inx 0 (1+ row-inx)))
((>= row-inx row-count) tri)
(let ((row (make-vector (1+ row-inx))))
(vector-set! tri row-inx row)
(do ((col-inx 1 (1+ col-inx)))
((>= col-inx (vector-length row)))
(vector-set! row col-inx
(* (vector-ref (vector-ref tri (1- row-inx)) (1- col-inx))
(/ row-inx (1+ col-inx)))))
(calc-store-first! row))))))
; Convert elements of a vector to a string for display.
(define vector->string
(lambda (vec)
(do ((inx 0 (1+ inx))
(str "" (string-append str (format "~7@a" (vector-ref vec inx)))))
((>= inx (vector-length vec)) str))))
; Display a Faulhaber's Triangle.
(define faulhabers-triangle-display
(lambda (tri)
(do ((inx 0 (1+ inx)))
((>= inx (vector-length tri)))
(printf "~a~%" (vector->string (vector-ref tri inx))))))
; Task..
(let ((row-count 10))
(printf "The first ~a rows of Faulhaber's Triangle..~%" row-count)
(faulhabers-triangle-display (faulhabers-triangle row-count)))
(newline)
(let ((power 17)
(sum-to 1000))
(printf "Sum over k=1..~a of k^~a using Faulhaber's Triangle..~%" sum-to power)
(let* ((tri (faulhabers-triangle (1+ power)))
(coefs (vector-ref tri power)))
(printf "~a~%" (do ((inx 0 (1+ inx))
(term-expt sum-to (* term-expt sum-to))
(sum 0 (+ sum (* (vector-ref coefs inx) term-expt))))
((>= inx (vector-length coefs)) sum)))))</lang>
{{out}}
<pre>
The first 10 rows of Faulhaber's Triangle..
1
1/2 1/2
1/6 1/2 1/3
0 1/4 1/2 1/4
-1/30 0 1/3 1/2 1/5
0 -1/12 0 5/12 1/2 1/6
1/42 0 -1/6 0 1/2 1/2 1/7
0 1/12 0 -7/24 0 7/12 1/2 1/8
-1/30 0 2/9 0 -7/15 0 2/3 1/2 1/9
0 -3/20 0 1/2 0 -7/10 0 3/4 1/2 1/10
Sum over k=1..1000 of k^17 using Faulhaber's Triangle..
56056972216555580111030077961944183400198333273050000
</pre>
=={{header|Sidef}}==
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