A328807 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

1, 1, 3, 1, 3, 8, 1, 3, 9, 20, 1, 3, 11, 27, 48, 1, 3, 15, 45, 81, 112, 1, 3, 23, 93, 195, 243, 256, 1, 3, 39, 225, 639, 873, 729, 576, 1, 3, 71, 597, 2583, 4653, 3989, 2187, 1280, 1, 3, 135, 1665, 11991, 32133, 35169, 18483, 6561, 2816

Offset

0,3

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 is 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

Links

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

Example

Square array begins:
     1,   1,   1,    1,     1,      1, ...
     3,   3,   3,    3,     3,      3, ...
     8,   9,  11,   15,    23,     39, ...
    20,  27,  45,   93,   225,    597, ...
    48,  81, 195,  639,  2583,  11991, ...
   112, 243, 873, 4653, 32133, 260613, ...

Mathematica

T[n_, k_] := Sum[Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

Crossrefs

Columns k=0..5 give A001792, A000244, A026375, A002893, A328808, A328809.

Main diagonal gives A328810.

Cf. A309010, A328747, A328748.

Sequence in context: A019603 A171843 A132476 * A103279 A208910 A209760

Adjacent sequences: A328804 A328805 A328806 * A328808 A328809 A328810

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 28 2019

Status

approved