gregory_coefficients.sf

#!/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("G(%2d) = %40s / %s\n", n, a(n) -> nude)
}

__END__
G( 0) =                                        1 / 1
G( 1) =                                        1 / 2
G( 2) =                                       -1 / 12
G( 3) =                                        1 / 24
G( 4) =                                      -19 / 720
G( 5) =                                        3 / 160
G( 6) =                                     -863 / 60480
G( 7) =                                      275 / 24192
G( 8) =                                   -33953 / 3628800
G( 9) =                                     8183 / 1036800
G(10) =                                 -3250433 / 479001600
G(11) =                                     4671 / 788480
G(12) =                             -13695779093 / 2615348736000
G(13) =                               2224234463 / 475517952000
G(14) =                            -132282840127 / 31384184832000
G(15) =                               2639651053 / 689762304000
G(16) =                         -111956703448001 / 32011868528640000
G(17) =                                 50188465 / 15613165568
G(18) =                        -2334028946344463 / 786014494949376000
G(19) =                          301124035185049 / 109285437800448000
G(20) =                    -12365722323469980029 / 4817145976189747200000
G(21) =                      8519318716801273673 / 3549475982455603200000
G(22) =                  -1232577428602510264423 / 547454472117564211200000
G(23) =                          530916160966849 / 250639102771200000
G(24) =             -101543126947618093900697699 / 50814724101952310083584000000
G(25) =                 439498633365840119748791 / 232561666370491121664000000
G(26) =              -64252172543850268483123097 / 35869217013142807117824000000
G(27) =                    928685729779901399375 / 545814099444746491527168
G(28) =         -1718089509598695642524656240811 / 1061011439248764234545233920000000
G(29) =                   5150046951561533494311 / 3335806532892753920000000
G(30) =     -44810233755305010150728029810063187 / 30391611665841602734313680404480000000