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

0,3

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A028313.

Sequence in context: A113811 A126168 A331970 * A137235 A021166 A131231

Adjacent sequences: A028320 A028321 A028322 * A028324 A028325 A028326

Keyword

nonn,easy,tabf

Author

Mohammad K. Azarian

Extensions

More terms from James A. Sellers

Status

approved