A028323 Elements to the right of the central elements of the 5-Pascal triangle A028313.

1, 1, 6, 1, 7, 1, 19, 8, 1, 27, 9, 1, 65, 36, 10, 1, 101, 46, 11, 1, 231, 147, 57, 12, 1, 378, 204, 69, 13, 1, 840, 582, 273, 82, 14, 1, 1422, 855, 355, 96, 15, 1, 3102, 2277, 1210, 451, 111, 16, 1, 5379, 3487, 1661, 562, 127, 17, 1, 11583, 8866, 5148, 2223, 689, 144, 18, 1

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy
• tabf

Examples

			This sequence represents the following portion of A028313(n,k), with x being the elements of A028313(2*n,n):
  x,
  .,  1,
  .,  x,  1,
  .,  .,  6,   1,
  .,  .,  x,   7,   1,
  .,  ., ..,  19,   8,   1,
  .,  ., ..,   x,  27,   9,   1,
  ., .., ..,  ..,  65,  36,  10,  1,
  ., .., .., ...,   x, 101,  46, 11,  1,
  ., .., .., ..., ..., 231, 147, 57, 12, 1.
As an irregular triangle this sequence begins as:
     1;
     1;
     6,    1;
     7,    1;
    19,    8,    1;
    27,    9,    1;
    65,   36,   10,   1;
   101,   46,   11,   1;
   231,  147,   57,  12,   1;
   378,  204,   69,  13,   1;
   840,  582,  273,  82,  14,  1;
  1422,  855,  355,  96,  15,  1;
  3102, 2277, 1210, 451, 111, 16,  1;
		

Links

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

Crossrefs

Cf. A028313.

Programs

Magma

A028323:= func< n,k | n eq 0 select 1 else Binomial(n+1, k + Floor((n+1)/2) + 1) + 3*Binomial(n-1, k + Floor((n+1)/2)) >;
[A028323(n,k): k in [0..Floor(n/2)], n in [0..16]]; // G. C. Greubel, Jan 05 2024

Mathematica

T[n_, k_]:= Binomial[n+1, k +Floor[(n+1)/2] +1] + 3*Binomial[n-1, k+ Floor[(n+1)/2]] -3*Boole[n==0];
Table[T[n,k], {n,0,16}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Jan 05 2024 *)

SageMath

def A028323(n,k): return binomial(n+1, k+1+(n+1)//2) + 3*binomial(n-1, k+((n+1)//2)) - 3*int(n==0)
flatten([[A028323(n,k) for k in range(1+(n//2))] for n in range(17)]) # G. C. Greubel, Jan 05 2024

Formula

From G. C. Greubel, Jan 05 2024: (Start)

a(n) = A028313(n, k), for 1 + floor(n/2) <= k <= n, n >= 0.

T(n, k) = binomial(n+1, k + floor((n+1)/2) + 1) + 3*binomial(n-1, k + floor((n+1)/2)) -3*[n=0], for 0 <= k <= floor(n/2), n >= 0. (End)

Extensions

More terms from James Sellers