Binomial transform

Generates the sequences on the fly.

sub binomial { [×] ($^n … 0) Z/ 1 .. $^p }

sub binomial-transform (*@seq) {
    @seq.keys.map: -> \n { sum (0..n).map: -> \k { binomial(n,k) × @seq[k] } }
}

sub inverse-binomial-transform (*@seq) {
    @seq.keys.map: -> \n { sum (0..n).map: -> \k { binomial(n,k) × @seq[k] × exp(n - k, -1) } }
}

sub si-binomial-transform (*@seq) { #self inverting
    @seq.keys.map: -> \n { sum (0..n).map: -> \k { binomial(n,k) × @seq[k] × exp(k, -1) } }
}

my $upto = 20;

for 'Catalan number',   (1, {[+] @_ Z× @_.reverse}…*),
    'Prime flip-flop',  (1..*).map({.is-prime ?? 1 !! 0}),
    'Fibonacci number', (0,1,*+*…*),
    'Padovan number',   (1,0,0, -> $c,$b,$ {$b+$c}…*)
  -> $name, @seq {
    say qq:to/BIN/;
    $name sequence:
    {@seq[^$upto]}
    Forward binomial transform:
    {binomial-transform(@seq)[^$upto]}
    Inverse binomial transform:
    {inverse-binomial-transform(@seq)[^$upto]}
    Round trip:
    {inverse-binomial-transform(binomial-transform(@seq))[^$upto]}
    Self inverting:
    {si-binomial-transform(@seq)[^$upto]}
    Re inverted:
    {si-binomial-transform(si-binomial-transform(@seq))[^$upto]}
    BIN
}

Output:

Catalan number sequence:
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190
Forward binomial transform:
1 2 5 15 51 188 731 2950 12235 51822 223191 974427 4302645 19181100 86211885 390248055 1777495635 8140539950 37463689775 173164232965
Inverse binomial transform:
1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595
Round trip:
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190
Self inverting:
1 0 1 -1 3 -6 15 -36 91 -232 603 -1585 4213 -11298 30537 -83097 227475 -625992 1730787 -4805595
Re inverted:
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190

Prime flip-flop sequence:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0
Forward binomial transform:
0 1 3 6 11 20 37 70 134 255 476 869 1564 2821 5201 9948 19793 40562 84271 174952
Inverse binomial transform:
0 1 -1 0 3 -10 25 -56 118 -237 456 -847 1540 -2795 5173 -9918 19761 -40528 84235 -174914
Round trip:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0
Self inverting:
0 -1 -1 0 3 10 25 56 118 237 456 847 1540 2795 5173 9918 19761 40528 84235 174914
Re inverted:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0

Fibonacci number sequence:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
Forward binomial transform:
0 1 3 8 21 55 144 377 987 2584 6765 17711 46368 121393 317811 832040 2178309 5702887 14930352 39088169
Inverse binomial transform:
0 1 -1 2 -3 5 -8 13 -21 34 -55 89 -144 233 -377 610 -987 1597 -2584 4181
Round trip:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
Self inverting:
0 -1 -1 -2 -3 -5 -8 -13 -21 -34 -55 -89 -144 -233 -377 -610 -987 -1597 -2584 -4181
Re inverted:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181

Padovan number sequence:
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37
Forward binomial transform:
1 1 1 2 5 12 28 65 151 351 816 1897 4410 10252 23833 55405 128801 299426 696081 1618192
Inverse binomial transform:
1 -1 1 0 -3 10 -24 49 -89 145 -208 245 -174 -176 1121 -3185 7137 -13920 24301 -37926
Round trip:
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37
Self inverting:
1 1 1 0 -3 -10 -24 -49 -89 -145 -208 -245 -174 176 1121 3185 7137 13920 24301 37926
Re inverted:
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37