Binomial transform

The binomial transform is a bijective sequence transform based on convolution with binomial coefficients.

It may be thought of as an nth forward difference with odd differences carrying a negative sign.

There are two common variants of binomial transforms, one of which is self-inverting; reapplying the transform to a transformed sequence returns the original sequence, and one which has separate "forward" and "inverse" transform operations.

The two variants only differ in placement and quantity of signs. The variant standardized on by OEIS, with the 'forward' and 'inverse' complementary operations, will be used here.

In this variant, to transform the sequence a to sequence b and back:

the forward binomial transform is defined as: $b_{n}=\sum _{k=0}^{n}{n \choose k}a_{k}.$

and the inverse binomial transform is defined as: $a_{n}=\sum _{k=0}^{n}(-1)^{n-k}{n \choose k}b_{k}$

where ${n \choose k}$ is the binomial operator 'n choose k'.

Task

Implement both a forward, and inverse binomial transform routine.
Use those routines to compute the forward binomial transform, the inverse binomial transform, and the inverse of the forward transform (should return original sequence) of a few representative sequences.
Show at least the first 15 values in each sequence.

You may generate the sequences, or may choose to just hard code the values.

Use the following sequences for testing:

Catalan numbers: 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440

Prime flip flop sequence: (for a_n: 1 if prime, 0 otherwise): 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0

Fibonacci sequence: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377

Padovan sequence: (starting with values 1,0,0): 1 0 0 1 0 1 1 1 2 2 3 4 5 7 9

See also

Wikipedia: Binomial transform OEIS wiki: Binomial transform

first sequence

OEIS:A007317 - Binomial transform of Catalan numbers OEIS:A005043 - Riordan numbers (Inverse binomial transform of Catalan numbers)

second sequence

OEIS:A052467 - Binomial transform of {b(n)}, where b(n)=1 for prime n and b(n)=0 otherwise (Binomial transform of Prime flip-flop sequence

third sequence

OEIS:A001906 - F(2n) = bisection of Fibonacci sequence (Binomial transform of Fibonacci sequence) OEIS:A039834 - a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with a(-2) = a(-1) = 1 (Inverse binomial transform of Fibonacci sequence)

fourth sequence

OEIS:A034943 - Binomial transform of Padovan sequence OEIS:A144413 - a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4) (Inverse binomial transform of Padovan sequence)

Raku

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 { exp(n - k, -1) × binomial(n,k) × @seq[k] } }
}

my $upto = 20;

for 'Catalan number',   (1, { [+] @_ Z× @_.reverse }…*),
    'Prime flip-flop',  (1..*).map({.is-prime ?? 1 !! 0}),
    'Fibonacci number', (0,1,1,*+*…*),
    '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]}
    Re-inverted:
    {inverse-binomial-transform(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
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
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
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
Re-inverted:
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37