blandin-diaz_compositional_bernoulli_numbers_B_Z_1_2.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 24 February 2018
# https://github.com/trizen

# A new recurrence for computing Blandin-Diaz compositional Bernoulli numbers (B^Z)_1,n.

# Formula:
#  a(0) = 1
#  a(n) = -(Sum_{k=0..n-1} a(k) * binomial(n+1, k) / (n-k+1)) / (n+1)

# Which gives us the nth Blandin-Diaz compositional Bernoulli number as:
#   (B^Z)_1,n = a(n)

# See also:
#   https://arxiv.org/abs/0708.0809

# OEIS entries:
#   https://oeis.org/A132096   (numerators)
#   https://oeis.org/A132097   (denominators)

func a((0)) { 1 }

func a(n) is cached {
    -sum(^n, {|k| a(k) * binomial(n+1, k) / (n - k + 1) }) / (n+1)
}

for n in (0..30) {
    printf("(B^Z)_1(%2d) = %50s / %s\n", n, a(n) -> nude)
}

__END__
(B^Z)_1( 0) =                                                  1 / 1
(B^Z)_1( 1) =                                                 -1 / 4
(B^Z)_1( 2) =                                                  1 / 72
(B^Z)_1( 3) =                                                  1 / 96
(B^Z)_1( 4) =                                                 61 / 21600
(B^Z)_1( 5) =                                                 -1 / 640
(B^Z)_1( 6) =                                             -12491 / 5080320
(B^Z)_1( 7) =                                               -479 / 580608
(B^Z)_1( 8) =                                             530629 / 326592000
(B^Z)_1( 9) =                                              54979 / 20736000
(B^Z)_1(10) =                                            1039405 / 2529128448
(B^Z)_1(11) =                                           -4981183 / 1094860800
(B^Z)_1(12) =                                     -9055875786121 / 1298164008960000
(B^Z)_1(13) =                                       908993573959 / 399435079680000
(B^Z)_1(14) =                                    288260975797477 / 11298306539520000
(B^Z)_1(15) =                                      7874837285353 / 231760134144000
(B^Z)_1(16) =                               -2255621632465386299 / 48978158848819200000
(B^Z)_1(17) =                                -189404901989770501 / 768284844687360000
(B^Z)_1(18) =                           -20038592583515962234111 / 81541143706048266240000
(B^Z)_1(19) =                              954329155426992424481 / 1009797445276139520000
(B^Z)_1(20) =                       1731149375200514221429374109 / 467359502609929273344000000
(B^Z)_1(21) =                          8016281400739796361657439 / 4685308296841396224000000
(B^Z)_1(22) =                   -1964113542866695843598946823499 / 77059691495268338368512000000
(B^Z)_1(23) =                    -118691634224452471169536026691 / 1489076164159774654464000000
(B^Z)_1(24) =               135434809276184922849073184461382701 / 4161725903949894195845529600000000
(B^Z)_1(25) =                26583901529200186483827848422605931 / 28951136723129698753708032000000
(B^Z)_1(26) =              2091948019281596823581094063056026741 / 921982354105822714156548096000000
(B^Z)_1(27) =               -17184038039286229015503662725836073 / 3902570811029937414419251200000
(B^Z)_1(28) =          -8419050697190151468948558716310571435121 / 194332621504510501906179686400000000
(B^Z)_1(29) =         -13029523224822054632420243355173393801041 / 169761830279802277527237427200000000
(B^Z)_1(30) =   277675025860127837282124790366392259270502106569 / 696429859645093494977344849204740096000000