Faulhaber's formula: Difference between revisions

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 =={{header|Sidef}}==
-<lang ruby>func faulhaber_s_formula(p) {
+<lang ruby>const AnyNum = require('Math::AnyNum')
+const Poly   = require('Math::Polynomial')
+Poly.string_config(Hash(
-    var formula = gather {
-        { |j|
+    fold_sign => true, prefix => "",
+    suffix => "", variable => "n"
-            take "(#{binomial(p+1, j) * j.bernfrac -> as_rat})*n^#{p+1 - j}"
+))
-        } << 0..p
-    }
+func anynum(n) {
-    formula.grep! { !.contains('(0)*') }.join!(' + ')
+    AnyNum.new(n.as_rat)
-    formula -= /\(1\)\*/g
-    formula -= /\^1\b/g
-    formula.gsub!(/\(([^+]*?)\)/, { _ })
-    "1/#{p + 1} * (#{formula})"
 }
+func faulhaber_formula(p) {
-{ |p|
+    (p+1).of { |j|
-    printf("%2d: %s\n", p, faulhaber_s_formula(p))
+        Poly.monomial(p - j + 1)\
-} << ^10</lang>
+            .mul_const(anynum(bernoulli(j)))\
-{{out}}
+            .mul_const(anynum(binomial(p+1, j)))
-<pre>
+    }.reduce(:add).div_const(p+1)
-: 1/1 * (n)
-: 1/2 * (n^2 + n)
-: 1/3 * (n^3 + 3/2*n^2 + 1/2*n)
-: 1/4 * (n^4 + 2*n^3 + n^2)
-: 1/5 * (n^5 + 5/2*n^4 + 5/3*n^3 + -1/6*n)
-: 1/6 * (n^6 + 3*n^5 + 5/2*n^4 + -1/2*n^2)
-: 1/7 * (n^7 + 7/2*n^6 + 7/2*n^5 + -7/6*n^3 + 1/6*n)
-: 1/8 * (n^8 + 4*n^7 + 14/3*n^6 + -7/3*n^4 + 2/3*n^2)
-: 1/9 * (n^9 + 9/2*n^8 + 6*n^7 + -21/5*n^5 + 2*n^3 + -3/10*n)
-: 1/10 * (n^10 + 5*n^9 + 15/2*n^8 + -7*n^6 + 5*n^4 + -3/2*n^2)
-</pre>
-By not simplifying the formulas, we can have a much cleaner code:
-<lang ruby>func faulhaber_s_formula(p) {
-    "1/#{p + 1} * (" + gather {
-      { |j|
-         take "#{binomial(p+1, j) * j.bernfrac -> as_rat}*n^#{p+1 - j}"
-      } << 0..p
-    }.join(' + ') + ")"
 }
+for p in (^10) {
-{ |p|
-    printf("%2d: %s\n", p, faulhaber_s_formula(p))
+    printf("%2d: %s\n", p, faulhaber_formula(p))
-} << ^10</lang>
+}</lang>
 {{out}}
 <pre>
-: 1/1 * (1*n^1)
+: n
-: 1/2 * (1*n^2 + 1*n^1)
+: 1/2 n^2 + 1/2 n
-: 1/3 * (1*n^3 + 3/2*n^2 + 1/2*n^1)
+: 1/3 n^3 + 1/2 n^2 + 1/6 n
-: 1/4 * (1*n^4 + 2*n^3 + 1*n^2 + 0*n^1)
+: 1/4 n^4 + 1/2 n^3 + 1/4 n^2
-: 1/5 * (1*n^5 + 5/2*n^4 + 5/3*n^3 + 0*n^2 + -1/6*n^1)
+: 1/5 n^5 + 1/2 n^4 + 1/3 n^3 - 1/30 n
-: 1/6 * (1*n^6 + 3*n^5 + 5/2*n^4 + 0*n^3 + -1/2*n^2 + 0*n^1)
+: 1/6 n^6 + 1/2 n^5 + 5/12 n^4 - 1/12 n^2
-: 1/7 * (1*n^7 + 7/2*n^6 + 7/2*n^5 + 0*n^4 + -7/6*n^3 + 0*n^2 + 1/6*n^1)
+: 1/7 n^7 + 1/2 n^6 + 1/2 n^5 - 1/6 n^3 + 1/42 n
-: 1/8 * (1*n^8 + 4*n^7 + 14/3*n^6 + 0*n^5 + -7/3*n^4 + 0*n^3 + 2/3*n^2 + 0*n^1)
+: 1/8 n^8 + 1/2 n^7 + 7/12 n^6 - 7/24 n^4 + 1/12 n^2
-: 1/9 * (1*n^9 + 9/2*n^8 + 6*n^7 + 0*n^6 + -21/5*n^5 + 0*n^4 + 2*n^3 + 0*n^2 + -3/10*n^1)
+: 1/9 n^9 + 1/2 n^8 + 2/3 n^7 - 7/15 n^5 + 2/9 n^3 - 1/30 n
-: 1/10 * (1*n^10 + 5*n^9 + 15/2*n^8 + 0*n^7 + -7*n^6 + 0*n^5 + 5*n^4 + 0*n^3 + -3/2*n^2 + 0*n^1)
+: 1/10 n^10 + 1/2 n^9 + 3/4 n^8 - 7/10 n^6 + 1/2 n^4 - 3/20 n^2
 </pre>