A307394 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k+x^k).

1, 1, 3, 1, 4, 6, 1, 4, 9, 10, 1, 4, 10, 14, 15, 1, 4, 10, 19, 15, 21, 1, 4, 10, 20, 28, 8, 28, 1, 4, 10, 20, 34, 28, -7, 36, 1, 4, 10, 20, 35, 48, 1, -22, 45, 1, 4, 10, 20, 35, 55, 48, -80, -21, 55, 1, 4, 10, 20, 35, 56, 75, 0, -242, 12, 66, 1, 4, 10, 20, 35, 56, 83, 75, -164, -485, 77, 78

Offset: 0

Author

Seiichi Manyama, Apr 07 2019

Keywords

• sign
• tabl

Examples

			Square array begins:
    1,   1,    1,    1,  1,   1,   1,   1,   1, ...
    3,   4,    4,    4,  4,   4,   4,   4,   4, ...
    6,   9,   10,   10, 10,  10,  10,  10,  10, ...
   10,  14,   19,   20, 20,  20,  20,  20,  20, ...
   15,  15,   28,   34, 35,  35,  35,  35,  35, ...
   21,   8,   28,   48, 55,  56,  56,  56,  56, ...
   28,  -7,    1,   48, 75,  83,  84,  84,  84, ...
   36, -22,  -80,    0, 75, 110, 119, 120, 120, ...
   45, -21, -242, -164,  0, 110, 154, 164, 165, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns 1-5 give A000217(n+1), A279230, A307395, A099589(n+3), A289388(n+3).

Cf. A306914, A307039, A307079, A307393.

Programs

Mathematica

T[n_, k_] := Sum[(-1)^j * Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

Formula

A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+3,k*j+3).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+1,k*j+1) * binomial(n-i+1,k*j+1).