A307665 A(n,k) = Sum_{j=0..floor(n/k)} binomial(2*n,k*j+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.
1, 1, 3, 1, 2, 11, 1, 2, 7, 42, 1, 2, 6, 26, 163, 1, 2, 6, 21, 99, 638, 1, 2, 6, 20, 78, 382, 2510, 1, 2, 6, 20, 71, 297, 1486, 9908, 1, 2, 6, 20, 70, 262, 1145, 5812, 39203, 1, 2, 6, 20, 70, 253, 990, 4447, 22819, 155382, 1, 2, 6, 20, 70, 252, 936, 3796, 17358, 89846, 616666
Offset: 0
Examples
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
3, 2, 2, 2, 2, 2, 2, 2, ...
11, 7, 6, 6, 6, 6, 6, 6, ...
42, 26, 21, 20, 20, 20, 20, 20, ...
163, 99, 78, 71, 70, 70, 70, 70, ...
638, 382, 297, 262, 253, 252, 252, 252, ...
2510, 1486, 1145, 990, 936, 925, 924, 924, ...
9908, 5812, 4447, 3796, 3523, 3446, 3433, 3432, ...
Programs
-
Mathematica
T[n_, k_] := Sum[Binomial[2*n, k*j + n], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 13 2021*)