A027948 Triangular array T read by rows: T(n,k) = t(n,2k+1) for 0 <= k <= n, T(n,n)=1, t given by A027926, n >= 0.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 7, 4, 1, 1, 3, 8, 14, 5, 1, 1, 3, 8, 20, 25, 6, 1, 1, 3, 8, 21, 46, 41, 7, 1, 1, 3, 8, 21, 54, 97, 63, 8, 1, 1, 3, 8, 21, 55, 133, 189, 92, 9, 1, 1, 3, 8, 21, 55, 143, 309, 344, 129, 10, 1, 1, 3, 8, 21, 55, 144, 364, 674, 591, 175, 11, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Examples

			Triangle begins with:
  1;
  1, 1;
  1, 2, 1;
  1, 3, 3,  1;
  1, 3, 7,  4,  1;
  1, 3, 8, 14,  5,  1;
  1, 3, 8, 20, 25,  6, 1;
  1, 3, 8, 21, 46, 41, 7, 1; ...
		

Links

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

Crossrefs

The row sums of this (slightly extended) bisection of the "Fibonacci array" A027926 are powers of 2, see A027935 for the other bisection.

Programs

GAP

T:= function(n,k)
    if k=n then return 1;
    else return Sum([0..n-k], j-> Binomial(n-j, 2*(n-k-j)-1) );
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Sep 29 2019

Magma

T:= func< n,k | k eq n select 1 else &+[Binomial(n-j, 2*(n-k-j) -1): j in [0..n-k]] >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2019

Maple

T:= proc(n, k)
      if k=n then 1
      else add(binomial(n-j, 2*(n-k-j)-1), j=0..n-k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Sep 29 2019

Mathematica

T[n_, k_]:= If[k==n, 1, Sum[Binomial[n-j, 2*(n-k-j)-1], {j, 0, n-k}]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 29 2019 *)

PARI

T(n,k) = if(k==n, 1, sum(j=0,n-k, binomial(n-j, 2*(n-k-j)-1)) );
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 29 2019

Sage

def T(n, k):
    if (k==n): return 1
    else: return sum(binomial(n-j, 2*(n-k-j)-1) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 29 2019

Formula

T(n,k) = Sum_{j=0..n-k} binomial(n-j, 2*(n-k-j) -1) with T(n,n)=1 in the region n >= 0, 0 <= k <= n. - G. C. Greubel, Sep 29 2019

Extensions

Name edited by G. C. Greubel, Sep 29 2019