Binomial transform: Difference between revisions

Content added Content deleted
m (Show self inverting variant for reference)
m (→‎{{header|Raku}}: Show self inverting variant also, other minor twiddles)
Line 487: Line 487:


sub inverse-binomial-transform (*@seq) {
sub inverse-binomial-transform (*@seq) {
@seq.keys.map: -> \n { sum (0..n).map: -> \k { exp(n - k, -1) × binomial(n,k) × @seq[k] } }
@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;
my $upto = 20;


for 'Catalan number', (1, { [+] @_ Z× @_.reverse }…*),
for 'Catalan number', (1, {[+] @_ Z× @_.reverse}…*),
'Prime flip-flop', (1..*).map({.is-prime ?? 1 !! 0}),
'Prime flip-flop', (1..*).grep(&is-prime).map(+*),
'Fibonacci number', (0,1,1,*+*…*),
'Fibonacci number', (0,1,*+*…*),
'Padovan number', (1,0,0, -> $c,$b,$ {$b+$c}…*)
'Padovan number', (1,0,0, -> $c,$b,$ {$b+$c}…*)
-> $name, @seq {
-> $name, @seq {
Line 506: Line 510:
Round trip:
Round trip:
{inverse-binomial-transform(binomial-transform(@seq))[^$upto]}
{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
BIN
}</syntaxhighlight>
}</syntaxhighlight>
{{out}}
{{out}}
<pre>Catalan number sequence:
<pre style="height:40ex;overflow:scroll;">Catalan number sequence:
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190
Forward binomial transform:
Forward binomial transform:
Line 516: Line 524:
1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595
1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595
Round trip:
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
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190


Prime flip-flop sequence:
Prime flip-flop sequence:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
Forward binomial transform:
Forward binomial transform:
2 5 13 33 83 205 495 1169 2707 6169 13889 30993 68701 151469 332349 725837 1577751 3413221 7349029 15751187
0 1 3 6 11 20 37 70 134 255 476 869 1564 2821 5201 9948 19793 40562 84271 174952
Inverse binomial transform:
Inverse binomial transform:
0 1 -1 0 3 -10 25 -56 118 -237 456 -847 1540 -2795 5173 -9918 19761 -40528 84235 -174914
2 1 1 -1 3 -9 23 -53 115 -237 457 -801 1213 -1389 445 3667 -15081 41335 -95059 195769
Round trip:
Round trip:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
Self inverting:
2 -1 1 1 3 9 23 53 115 237 457 801 1213 1389 445 -3667 -15081 -41335 -95059 -195769
Re inverted:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71


Fibonacci number sequence:
Fibonacci number sequence:
Line 534: Line 550:
0 1 -1 2 -3 5 -8 13 -21 34 -55 89 -144 233 -377 610 -987 1597 -2584 4181
0 1 -1 2 -3 5 -8 13 -21 34 -55 89 -144 233 -377 610 -987 1597 -2584 4181
Round trip:
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
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181


Line 543: Line 563:
1 -1 1 0 -3 10 -24 49 -89 145 -208 245 -174 -176 1121 -3185 7137 -13920 24301 -37926
1 -1 1 0 -3 10 -24 49 -89 145 -208 245 -174 -176 1121 -3185 7137 -13920 24301 -37926
Round trip:
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</pre>
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37</pre>


Retrieved from "https://rosettacode.org/wiki/Binomial_transform"