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A328748
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.
7
1, 1, 0, 1, 0, -1, 1, 0, 0, 2, 1, 0, 2, 0, -3, 1, 0, 6, 0, 0, 4, 1, 0, 14, 12, 6, 0, -5, 1, 0, 30, 72, 90, 0, 0, 6, 1, 0, 62, 300, 882, 360, 20, 0, -7, 1, 0, 126, 1080, 6690, 8400, 2040, 0, 0, 8, 1, 0, 254, 3612, 44706, 124920, 95180, 10080, 70, 0, -9
OFFSET
0,10
COMMENTS
T(n,k) is the constant term in the expansion of (-2 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.
LINKS
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 0, 0, 0, 0, 0, ...
-1, 0, 2, 6, 14, 30, ...
2, 0, 0, 12, 72, 300, ...
-3, 0, 6, 90, 882, 6690, ...
4, 0, 0, 360, 8400, 124920, ...
MATHEMATICA
T[n_, k_] := Sum[(-2)^(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 *)
CROSSREFS
Columns k=0..5 give A097141(n+1), A000007, A126869, A002898, A328735, A328751.
T(n,n+1) gives A328814.
Sequence in context: A351977 A097567 A022881 * A093201 A067613 A264034
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Oct 27 2019
STATUS
approved