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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. 8

%I #40 May 06 2021 03:15:59

%S 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 115,175,51,1,0,1,1,63,391,1255,991,141,1,0,1,1,127,1267,8071,13671,

%U 5881,393,1,0,1,1,255,3991,49399,161671,160461,35617,1107,1,0

%N 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 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, 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="/A328747/b328747.txt">Antidiagonals n = 0..100, flattened</a>

%e Square array begins:

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

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

%e 0, 1, 3, 7, 15, 31, ...

%e 0, 1, 7, 31, 115, 391, ...

%e 0, 1, 19, 175, 1255, 8071, ...

%e 0, 1, 51, 991, 13671, 161671, ...

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

%Y Main diagonal gives A328811.

%Y T(n,n+1) gives A328813.

%Y Cf. A309010, A328748, A328807.

%K nonn,tabl

%O 0,13

%A _Seiichi Manyama_, Oct 27 2019

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)