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A307039
Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k+x^k).
10
1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, -2, 0, 1, 1, 1, 0, -4, 0, 1, 1, 1, 1, -3, -4, 0, 1, 1, 1, 1, 0, -9, 0, 0, 1, 1, 1, 1, 1, -4, -18, 8, 0, 1, 1, 1, 1, 1, 0, -14, -27, 16, 0, 1, 1, 1, 1, 1, 1, -5, -34, -27, 16, 0, 1, 1, 1, 1, 1, 1, 0, -20, -68, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, -6, -55, -116, 81, -32, 0
OFFSET
0,14
LINKS
FORMULA
A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n,k*j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 0, 1, 1, 1, 1, 1, 1, ...
0, -2, 0, 1, 1, 1, 1, 1, ...
0, -4, -3, 0, 1, 1, 1, 1, ...
0, 0, -18, -14, -5, 0, 1, 1, ...
0, 8, -27, -34, -20, -6, 0, 1, ...
0, 16, -27, -68, -55, -27, -7, 0, ...
0, 16, 0, -116, -125, -83, -35, -8, ...
MATHEMATICA
T[n_, k_] := Sum[(-1)^j * Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 13}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)
CROSSREFS
KEYWORD
sign,tabl,look
AUTHOR
Seiichi Manyama, Mar 21 2019
STATUS
approved