A328748 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

1, 1, 0, 1, 0, -1, 1, 0, 0, 2, 1, 0, 2, 0, -3, 1, 0, 6, 0, 0, 4, 1, 0, 14, 12, 6, 0, -5, 1, 0, 30, 72, 90, 0, 0, 6, 1, 0, 62, 300, 882, 360, 20, 0, -7, 1, 0, 126, 1080, 6690, 8400, 2040, 0, 0, 8, 1, 0, 254, 3612, 44706, 124920, 95180, 10080, 70, 0, -9

Offset: 0

Author

Seiichi Manyama, Oct 27 2019

Keywords

• sign
• tabl

Comments

T(n,k) is the constant term in the expansion of (-2 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.

Examples

			Square array begins:
    1, 1, 1,   1,    1,      1, ...
    0, 0, 0,   0,    0,      0, ...
   -1, 0, 2,   6,   14,     30, ...
    2, 0, 0,  12,   72,    300, ...
   -3, 0, 6,  90,  882,   6690, ...
    4, 0, 0, 360, 8400, 124920, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..100, flattened

Crossrefs

Columns k=0..5 give A097141(n+1), A000007, A126869, A002898, A328735, A328751.

T(n,n+1) gives A328814.

Cf. A309010, A328747, A328807.

Programs

Mathematica

T[n_, k_] := Sum[(-2)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)