new file: Math/gregory_coefficients.sf

Author: trizen

Math/gregory_coefficients.sf

@@ -0,0 +1,61 @@
+#!/usr/bin/ruby
+
+# Author: Daniel "Trizen" Șuteu
+# Date: 24 February 2018
+# https://github.com/trizen
+
+# A new recurrence for computing the logarithmic numbers (also known as Gregory coefficients).
+
+# Formula:
+#  a(0) = 1
+#  a(n) = Sum_{k=0..n-1} (-1)^(n - k + 1) * a(k) / (n - k + 1)
+
+# See also:
+#   https://oeis.org/A002206    (numerators)
+#   https://oeis.org/A002207    (denominators)
+
+# Wikipedia:
+#   https://en.wikipedia.org/wiki/Gregory_coefficients
+
+func a((0)) { 1 }
+
+func a(n) is cached {
+    sum(^n, {|k| (-1)**(n - k + 1) * a(k) / (n - k + 1) })
+}
+
+for n in (0..30) {
+    printf("L(%2d) = %40s / %s\n", n, a(n) -> nude)
+}
+
+__END__
+L( 0) =                                        1 / 1
+L( 1) =                                        1 / 2
+L( 2) =                                       -1 / 12
+L( 3) =                                        1 / 24
+L( 4) =                                      -19 / 720
+L( 5) =                                        3 / 160
+L( 6) =                                     -863 / 60480
+L( 7) =                                      275 / 24192
+L( 8) =                                   -33953 / 3628800
+L( 9) =                                     8183 / 1036800
+L(10) =                                 -3250433 / 479001600
+L(11) =                                     4671 / 788480
+L(12) =                             -13695779093 / 2615348736000
+L(13) =                               2224234463 / 475517952000
+L(14) =                            -132282840127 / 31384184832000
+L(15) =                               2639651053 / 689762304000
+L(16) =                         -111956703448001 / 32011868528640000
+L(17) =                                 50188465 / 15613165568
+L(18) =                        -2334028946344463 / 786014494949376000
+L(19) =                          301124035185049 / 109285437800448000
+L(20) =                    -12365722323469980029 / 4817145976189747200000
+L(21) =                      8519318716801273673 / 3549475982455603200000
+L(22) =                  -1232577428602510264423 / 547454472117564211200000
+L(23) =                          530916160966849 / 250639102771200000
+L(24) =             -101543126947618093900697699 / 50814724101952310083584000000
+L(25) =                 439498633365840119748791 / 232561666370491121664000000
+L(26) =              -64252172543850268483123097 / 35869217013142807117824000000
+L(27) =                    928685729779901399375 / 545814099444746491527168
+L(28) =         -1718089509598695642524656240811 / 1061011439248764234545233920000000
+L(29) =                   5150046951561533494311 / 3335806532892753920000000
+L(30) =     -44810233755305010150728029810063187 / 30391611665841602734313680404480000000

Math/motzkin_triangle.sf

@@ -55,6 +55,6 @@ __END__
    1   10   54  200  560 1242 2235 3288 3915 3610 2188
    1   11   65  264  814 2002 4037 6765 9438 10813 9713 5798
 
-=> Sum of the row of Motzkin's triangle:
+=> Sum of the rows of Motzkin's triangle:
 [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201917]
 [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201917]

README.md

@@ -105,6 +105,7 @@ A simple collection of Sidef scripts.
     * [Generalized continued fraction parts iter](./Math/generalized_continued_fraction_parts_iter.sf)
     * [Generalized fibonacci closed-form](./Math/generalized_fibonacci_closed-form.sf)
     * [Generalized fibonacci closed-form 2](./Math/generalized_fibonacci_closed-form_2.sf)
+    * [Gregory coefficients](./Math/gregory_coefficients.sf)
     * [Harmonic numbers from powers](./Math/harmonic_numbers_from_powers.sf)
     * [Harmonic numbers from powers simple](./Math/harmonic_numbers_from_powers_simple.sf)
     * [Harmonic prime powers](./Math/harmonic_prime_powers.sf)