OFFSET
0,6
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Number triangle T(n, k) = Sum_{j=0..floor(k/2)} binomial(n-j, j) if k <= n, 0 otherwise.
T(n, n) = A000045(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A099574(n).
Sum_{k=0..n} T(n, k) = A029907(n+1).
Antidiagonals of the following array: the first row equals the Fibonacci numbers, (1, 1, 2, 3, 5, ...), and the (n+1)-st row is obtained by the matrix-vector product A128174 * n-th row. - Gary W. Adamson, Jan 19 2011
From G. C. Greubel, Jul 25 2022: (Start)
T(n, n-1) = A052952(n-1), n >= 1.
T(n, n-2) = A054451(n-2), n >= 2.
T(n, n-3) = A099571(n-3), n >= 3.
T(n, n-4) = A099572(n-4), n >= 4. (End)
EXAMPLE
First few rows of the array:
1, 1, 2, 3, 5, 8, ... (A000045)
1, 1, 3, 4, 8, 12, ... (A052952)
1, 1, 4, 5, 12, 17, ... (A054451)
1, 1, 5, 6, 17, 23, ... (A099571)
1, 1, 6, 7, 23, 30, ... (A099572)
...
Triangle begins as:
1;
1, 1;
1, 1, 2;
1, 1, 3, 3;
1, 1, 4, 4, 5;
1, 1, 5, 5, 8, 8;
1, 1, 6, 6, 12, 12, 13;
1, 1, 7, 7, 17, 17, 21, 21;
1, 1, 8, 8, 23, 23, 33, 33, 34;
1, 1, 9, 9, 30, 30, 50, 50, 55, 55;
MATHEMATICA
T[n_, k_]:= Sum[Binomial[n-j, j], {j, 0, Floor[k/2]}];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)
PROG
(Magma) [(&+[Binomial(n-j, j): j in [0..Floor(k/2)]]): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 25 2022
(SageMath)
def A099573(n, k): return sum(binomial(n-j, j) for j in (0..(k//2)))
flatten([[A099573(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Oct 23 2004
STATUS
approved