OFFSET
0,1
LINKS
G. C. Greubel, Rows n = 0..100 of the irregular triangle, flattened
FORMULA
From G. C. Greubel, Jul 14 2024: (Start)
T(n, k) = 2*binomial(n+3, k+2 + floor((n+1)/2)).
Sum_{k=0..floor(n/2)} T(n, k) = A272514(n+3).
Sum_{k=0..n} (-1)^k*T(2*n, k) = 2*A286033(n+2).
Sum_{k=0..n} (-1)^k*T(2*n+1, k) = binomial(2*n+4, n+2) + 2*(-1)^n.
(End)
EXAMPLE
This sequence represents the following portion of A028330(n,k), with x being the elements of A028329(n):
x;
., .;
., x, .;
., ., 6, .;
., ., x, 8, .;
., ., ., 20, 10, .;
., ., ., x, 30, 12, .;
., ., ., ., 70, 42, 14, .;
., ., ., ., x, 112, 56, 16, .;
., ., ., ., ., 252, 168, 72, 18, .;
., ., ., ., ., x, 420, 240, 90, 20, .;
., ., ., ., ., ., 924, 660, 330, 110, 22, .;
., ., ., ., ., ., x, 1584, 990, 440, 132, 24, .;
As an irregular triangle:
6;
8;
20, 10;
30, 12;
70, 42, 14;
112, 56, 16;
252, 168, 72, 18;
420, 240, 90, 20;
924, 660, 330, 110, 22;
MATHEMATICA
Table[2*Binomial[n+3, k+2 +Floor[(n+1)/2]], {n, 0, 12}, {k, 0, Floor[n/2] }]//Flatten (* G. C. Greubel, Jul 14 2024 *)
PROG
(Magma)
[2*Binomial(n+3, k): k in [Floor((n+5)/2)..n+2], n in [0..12]]; // G. C. Greubel, Jul 14 2024
(SageMath)
def A028326(n, k): return 2*binomial(n, k)
flatten([[A028326(n+1, k) for k in range(((n+3)//2), n+1)] for n in range(21)]) # G. C. Greubel, Jul 14 2024
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
EXTENSIONS
More terms from James A. Sellers
STATUS
approved