A174093 Triangle T(n, k) = binomial(n-k+1, k) + binomial(k+1, n-k) with T(0,0) = T(1, 0) = T(1, 1) = 1, read by rows.

1, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 4, 6, 4, 1, 1, 5, 7, 7, 5, 1, 1, 6, 10, 8, 10, 6, 1, 1, 7, 15, 11, 11, 15, 7, 1, 1, 8, 21, 20, 10, 20, 21, 8, 1, 1, 9, 28, 35, 16, 16, 35, 28, 9, 1, 1, 10, 36, 56, 35, 12, 35, 56, 36, 10, 1

Offset: 0

Author

Roger L. Bagula, Mar 07 2010

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  4,  1;
  1,  4,  4,  1;
  1,  4,  6,  4,  1;
  1,  5,  7,  7,  5,  1;
  1,  6, 10,  8, 10,  6,  1;
  1,  7, 15, 11, 11, 15,  7,  1;
  1,  8, 21, 20, 10, 20, 21,  8,  1;
  1,  9, 28, 35, 16, 16, 35, 28,  9,  1;
  1, 10, 36, 56, 35, 12, 35, 56, 36, 10, 1;
		

Links

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

Crossrefs

Cf. A000045, A007318, A071679.

Cf. A174095, A174096, A174097.

Programs

Magma

T:= func< n,k | n lt 2 select 1 else Binomial(n-k+1, k) + Binomial(k+1, n-k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2021

Mathematica

T[n_, k_]:= If[n==0 || n==1, 1, Binomial[n-k+1, k] + Binomial[k+1, n-k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

Sage

def T(n, k):
    if (n==0 or n==1): return 1
    else: return binomial(n-k+1, k) + binomial(k+1, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 10 2021

Formula

T(n, k) = binomial(n-k+1, k) + binomial(k+1, n-k) with T(0,0) = T(1, 0) = T(1, 1) = 1.

From G. C. Greubel, Feb 10 2021: (Start)

T(n, k) = A007318(n-k+1, k) + A007318(k+1, n-k), for rows n >= 2.

T(n, k) = T(n, n-k).

Sum_{k=0..n} T(n, k) = 2*Fibonacci(n+2) - [n=0] - 2*[n=1] = 2*A071679(n) + [n=0], where [] is the Iverson bracket. (End)

Extensions

Edited by G. C. Greubel, Feb 10 2021