Binomial transform: Difference between revisions

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




=={{header|C}}==
=={{header|C}}==
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for 'Catalan number', (1, {[+] @_ Z× @_.reverse}…*),
for 'Catalan number', (1, {[+] @_ Z× @_.reverse}…*),
'Prime flip-flop', (1..*).grep(&is-prime).map(+*),
'Prime flip-flop', (1..*).map({.is-prime ?? 1 !! 0}),
'Fibonacci number', (0,1,*+*…*),
'Fibonacci number', (0,1,*+*…*),
'Padovan number', (1,0,0, -> $c,$b,$ {$b+$c}…*)
'Padovan number', (1,0,0, -> $c,$b,$ {$b+$c}…*)
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Prime flip-flop sequence:
Prime flip-flop sequence:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0
Forward binomial transform:
Forward binomial transform:
0 1 3 6 11 20 37 70 134 255 476 869 1564 2821 5201 9948 19793 40562 84271 174952
2 5 13 33 83 205 495 1169 2707 6169 13889 30993 68701 151469 332349 725837 1577751 3413221 7349029 15751187
Inverse binomial transform:
Inverse binomial transform:
2 1 1 -1 3 -9 23 -53 115 -237 457 -801 1213 -1389 445 3667 -15081 41335 -95059 195769
0 1 -1 0 3 -10 25 -56 118 -237 456 -847 1540 -2795 5173 -9918 19761 -40528 84235 -174914
Round trip:
Round trip:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0
Self inverting:
Self inverting:
2 -1 1 1 3 9 23 53 115 237 457 801 1213 1389 445 -3667 -15081 -41335 -95059 -195769
0 -1 -1 0 3 10 25 56 118 237 456 847 1540 2795 5173 9918 19761 40528 84235 174914
Re inverted:
Re inverted:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0


Fibonacci number sequence:
Fibonacci number sequence:
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