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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.

%I A328747 #40 May 06 2021 03:15:59
%S A328747 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,
%T A328747 115,175,51,1,0,1,1,63,391,1255,991,141,1,0,1,1,127,1267,8071,13671,
%U A328747 5881,393,1,0,1,1,255,3991,49399,161671,160461,35617,1107,1,0
%N 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.
%C A328747 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 A328747 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
%H A328747 Seiichi Manyama, <a href="/A328747/b328747.txt">Antidiagonals n = 0..100, flattened</a>
%e A328747 Square array begins:
%e A328747    1, 1,  1,   1,     1,      1, ...
%e A328747    1, 1,  1,   1,     1,      1, ...
%e A328747    0, 1,  3,   7,    15,     31, ...
%e A328747    0, 1,  7,  31,   115,    391, ...
%e A328747    0, 1, 19, 175,  1255,   8071, ...
%e A328747    0, 1, 51, 991, 13671, 161671, ...
%t A328747 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 *)
%Y A328747 Columns k=0..5 give A019590(n+1), A000012, A002426, A172634, A328725, A328750.
%Y A328747 Main diagonal gives A328811.
%Y A328747 T(n,n+1) gives A328813.
%Y A328747 Cf. A309010, A328748, A328807.
%K A328747 nonn,tabl
%O A328747 0,13
%A A328747 _Seiichi Manyama_, Oct 27 2019