A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 7, 1, 0, 1, 1, 15, 31, 19, 1, 0, 1, 1, 31, 115, 175, 51, 1, 0, 1, 1, 63, 391, 1255, 991, 141, 1, 0, 1, 1, 127, 1267, 8071, 13671, 5881, 393, 1, 0, 1, 1, 255, 3991, 49399, 161671, 160461, 35617, 1107, 1, 0

Offset: 0

Author

Seiichi Manyama, Oct 27 2019

Keywords

• nonn
• tabl

Comments

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

For fixed k > 0, T(n,k) ~ (2^k - 1)^(n + (k-1)/2) / (2^((k-1)^2/2) * sqrt(k) * (Pi*n)^((k-1)/2)). - Vaclav Kotesovec, Oct 28 2019

Examples

			Square array begins:
   1, 1,  1,   1,     1,      1, ...
   1, 1,  1,   1,     1,      1, ...
   0, 1,  3,   7,    15,     31, ...
   0, 1,  7,  31,   115,    391, ...
   0, 1, 19, 175,  1255,   8071, ...
   0, 1, 51, 991, 13671, 161671, ...
		

Links

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

Crossrefs

Columns k=0..5 give A019590(n+1), A000012, A002426, A172634, A328725, A328750.

Main diagonal gives A328811.

T(n,n+1) gives A328813.

Cf. A309010, A328748, A328807.

Programs

Mathematica

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