A106198 Triangle, columns = successive binomial transforms of Fibonacci numbers.

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, 34, 610, 1625, 1541, 757, 233, 50, 8, 1

Offset

0,4

Comments

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 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

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

Cf. A000045, A081567, A081568, A081569, A081570.

Sequence in context: A037027 A182810 A139375 * A202847 A054336 A342908

Adjacent sequences: A106195 A106196 A106197 * A106199 A106200 A106201

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 24 2005

Status

approved