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
KEYWORD
nonn,easy,tabf
AUTHOR
EXTENSIONS
More terms from James A. Sellers
STATUS
approved