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

Author

Gary W. Adamson, Apr 24 2005

Keywords

• nonn
• tabl

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.

Examples

			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, ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

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

Programs

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

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

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 *)

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

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

Formula

Offset column k = k-th binomial transform of the Fibonacci numbers, given leftmost column = Fibonacci numbers.