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

0,5

Links

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

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)

Example

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;

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

Prog

(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
(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

Crossrefs

Cf. A000045, A007318, A071679.

Cf. A174095, A174096, A174097.

Sequence in context: A053239 A046569 A046596 * A204028 A106314 A152716

Adjacent sequences: A174090 A174091 A174092 * A174094 A174095 A174096

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Mar 07 2010

Extensions

Edited by G. C. Greubel, Feb 10 2021

Status

approved