A328807 - OEIS

%I #25 May 06 2021 03:16:16

%S 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,

%T 256,1,3,39,225,639,873,729,576,1,3,71,597,2583,4653,3989,2187,1280,1,

%U 3,135,1665,11991,32133,35169,18483,6561,2816

%N 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.

%C 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.

%C 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

%H Seiichi Manyama, <a href="/A328807/b328807.txt">Antidiagonals n = 0..100, flattened</a>

%e Square array begins:

%e      1,   1,   1,    1,     1,      1, ...

%e      3,   3,   3,    3,     3,      3, ...

%e      8,   9,  11,   15,    23,     39, ...

%e     20,  27,  45,   93,   225,    597, ...

%e     48,  81, 195,  639,  2583,  11991, ...

%e    112, 243, 873, 4653, 32133, 260613, ...

%t 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 *)

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

%Y Main diagonal gives A328810.

%Y Cf. A309010, A328747, A328748.

%K nonn,tabl

%O 0,3

%A _Seiichi Manyama_, Oct 28 2019