%I #12 Jan 06 2024 09:22:47

%S 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,

%T 204,69,13,1,840,582,273,82,14,1,1422,855,355,96,15,1,3102,2277,1210,

%U 451,111,16,1,5379,3487,1661,562,127,17,1,11583,8866,5148,2223,689,144,18,1

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

%H G. C. Greubel, <a href="/A028323/b028323.txt">Rows n = 0..100 of the irregular triangle, flattened</a>

%F From _G. C. Greubel_, Jan 05 2024: (Start)

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

%F 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)

%e This sequence represents the following portion of A028313(n,k), with x being the elements of A028313(2*n,n):

%e x,

%e ., 1,

%e ., x, 1,

%e ., ., 6, 1,

%e ., ., x, 7, 1,

%e ., ., .., 19, 8, 1,

%e ., ., .., x, 27, 9, 1,

%e ., .., .., .., 65, 36, 10, 1,

%e ., .., .., ..., x, 101, 46, 11, 1,

%e ., .., .., ..., ..., 231, 147, 57, 12, 1.

%e As an irregular triangle this sequence begins as:

%e 1;

%e 1;

%e 6, 1;

%e 7, 1;

%e 19, 8, 1;

%e 27, 9, 1;

%e 65, 36, 10, 1;

%e 101, 46, 11, 1;

%e 231, 147, 57, 12, 1;

%e 378, 204, 69, 13, 1;

%e 840, 582, 273, 82, 14, 1;

%e 1422, 855, 355, 96, 15, 1;

%e 3102, 2277, 1210, 451, 111, 16, 1;

%t 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];

%t Table[T[n,k], {n,0,16}, {k,0,Floor[n/2]}]//Flatten (* _G. C. Greubel_, Jan 05 2024 *)

%o (Magma)

%o 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)) >;

%o [A028323(n,k): k in [0..Floor(n/2)], n in [0..16]]; // _G. C. Greubel_, Jan 05 2024

%o (SageMath)

%o 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)

%o flatten([[A028323(n,k) for k in range(1+(n//2))] for n in range(17)]) # _G. C. Greubel_, Jan 05 2024

%Y Cf. A028313.

%K nonn,easy,tabf

%O 0,3

%A _Mohammad K. Azarian_

%E More terms from _James A. Sellers_