OFFSET
0,4
COMMENTS
Column 0 = Fibonacci numbers, column 1 = odd-indexed Fibonacci numbers (first binomial transform of 1, 1, 2, 3, 5, ...); column 2 = second binomial transform of Fibonacci numbers, etc.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
Offset column k = k-th binomial transform of the Fibonacci numbers, given leftmost column = Fibonacci numbers.
EXAMPLE
First few rows of the triangle are:
1;
1, 1;
2, 2, 1;
3, 5, 3, 1;
5, 13, 10, 4, 1;
8, 34, 35, 17, 5, 1;
13, 89, 125, 75, 26, 6, 1;
21, 233, 450, 338, 139, 37, 7, 1;
...
Column 2 = A081567, second binomial transform of Fibonacci numbers: 1, 3, 10, 35, 125, ...
MAPLE
with(combinat);
T:= proc(n, k) option remember;
if k=0 then fibonacci(n+1)
else add( binomial(n-k, j)*fibonacci(j+1)*k^(n-k-j), j=0..n-k)
fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Dec 11 2019
MATHEMATICA
Table[If[k==0, Fibonacci[n+1], Sum[Binomial[n-k, j]*Fibonacci[j+1]*k^(n-k-j), {j, 0, n-k}]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 11 2019 *)
PROG
(PARI) T(n, k) = if(k==0, fibonacci(n+1), sum(j=0, n-k, binomial(n-k, j)*fibonacci( j+1)*k^(n-k-j)) ); \\ G. C. Greubel, Dec 11 2019
(Magma)
function T(n, k)
if k eq 0 then return Fibonacci(n+1);
else return (&+[Binomial(n-k, j)*Fibonacci(j+1)*k^(n-k-j): j in [0..n-k]]);
end if; return T; end function;
[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 11 2019
(Sage)
@CachedFunction
def T(n, k):
if (k==0): return fibonacci(n+1)
else: return sum(binomial(n-k, j)*fibonacci(j+1)*k^(n-k-j) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 11 2019
(GAP)
T:= function(n, k)
if k=0 then return Fibonacci(n+1);
else return Sum([0..n-k], j-> Binomial(n-k, j)*Fibonacci(j+1)*k^(n-k-j));
fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Dec 11 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Apr 24 2005
STATUS
approved