A307668 A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j*binomial(2*n,k*j+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.

1, 1, 1, 1, 2, 3, 1, 2, 5, 10, 1, 2, 6, 14, 35, 1, 2, 6, 19, 43, 126, 1, 2, 6, 20, 62, 142, 462, 1, 2, 6, 20, 69, 207, 494, 1716, 1, 2, 6, 20, 70, 242, 705, 1780, 6435, 1, 2, 6, 20, 70, 251, 858, 2445, 6563, 24310, 1, 2, 6, 20, 70, 252, 912, 3068, 8622, 24566, 92378

Offset: 0

Author

Seiichi Manyama, Apr 20 2019

Keywords

• nonn
• tabl

Examples

			Square array begins:
      1,    1,    1,     1,     1,     1,     1, ...
      1,    2,    2,     2,     2,     2,     2, ...
      3,    5,    6,     6,     6,     6,     6, ...
     10,   14,   19,    20,    20,    20,    20, ...
     35,   43,   62,    69,    70,    70,    70, ...
    126,  142,  207,   242,   251,   252,   252, ...
    462,  494,  705,   858,   912,   923,   924, ...
   1716, 1780, 2445,  3068,  3341,  3418,  3431, ...
   6435, 6563, 8622, 11051, 12310, 12750, 12854, ...
		

Crossrefs

Columns 1-2 give A088218, A005317.

Cf. A306914, A307039, A307394, A307665.

Programs

Mathematica

T[n_, k_] := Sum[(-1)^j*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*)