A027935 Triangular array T read by rows: T(n,k)=t(n,2k), t given by A027926; 0 <= k <= n, n >= 0.

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 7, 1, 1, 2, 5, 12, 11, 1, 1, 2, 5, 13, 26, 16, 1, 1, 2, 5, 13, 33, 51, 22, 1, 1, 2, 5, 13, 34, 79, 92, 29, 1, 1, 2, 5, 13, 34, 88, 176, 155, 37, 1, 1, 2, 5, 13, 34, 89, 221, 365, 247, 46, 1, 1, 2, 5, 13, 34, 89, 232, 530, 709, 376, 56, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 4,  1:
  1, 2, 5,  7,  1;
  1, 2, 5, 12, 11,  1;
  1, 2, 5, 13, 26, 16, 1;
  ...
		

Links

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

Crossrefs

The row sums of this bisection of the "Fibonacci array" A027926 are powers of 2, see A027948 for the other bisection.

Programs

GAP

Flat(List([0..12], n-> List([0..n], k-> Sum([0..Int((2*n-2*k+1)/2) ], j-> Binomial(n-j, 2*(n-k-j))) ))); # G. C. Greubel, Sep 27 2019

Magma

T:= func< n,k | &+[Binomial(n-j, 2*(n-k-j)) : j in [0..Floor((2*n -2*k+1)/2)]] >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2019

Maple

T:= proc(n, k) option remember;
     add( binomial(n-j, 2*(n-k-j)), j=0..floor((2*n - 2*k+1)/2))
   end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Sep 27 2019

Mathematica

T[n_, k_]:= Sum[Binomial[n-j, 2*(n-k-j)], {j,0,Floor[(2*n-2*k+1)/2]}]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 27 2019 *)

PARI

T(n,k) = sum(j=0,(2*n-2*k+1)\2, binomial(n-j, 2*(n-k-j)));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 27 2019

Sage

[[ sum(binomial(n-j, 2*(n-k-j)) for j in (0..floor((2*n-2*k+1)/2)) ) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 27 2019

Formula

T(n,k) = Sum_{j=0..floor((2*n-2*k-1)/2)} binomial(n-j, 2*(n-k-j)). - G. C. Greubel, Sep 27 2019