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A306913
Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k+x^k).
4
1, 1, -2, 1, -2, 4, 1, -3, 2, -8, 1, -4, 6, 0, 16, 1, -5, 10, -11, -4, -32, 1, -6, 15, -20, 21, 8, 64, 1, -7, 21, -35, 34, -42, -8, -128, 1, -8, 28, -56, 70, -48, 85, 0, 256, 1, -9, 36, -84, 126, -127, 48, -171, 16, -512, 1, -10, 45, -120, 210, -252, 220, 0, 342, -32, 1024
OFFSET
0,3
LINKS
FORMULA
A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^(((k+1) mod 2) * j) * binomial(n+k-1,k*j+k-1).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
-2, -2, -3, -4, -5, -6, -7, -8, ...
4, 2, 6, 10, 15, 21, 28, 36, ...
-8, 0, -11, -20, -35, -56, -84, -120, ...
16, -4, 21, 34, 70, 126, 210, 330, ...
-32, 8, -42, -48, -127, -252, -462, -792, ...
64, -8, 85, 48, 220, 461, 924, 1716, ...
-128, 0, -171, 0, -385, -780, -1717, -3432, ...
256, 16, 342, -164, 715, 1209, 3017, 6434, ...
MATHEMATICA
A[n_, k_] := (-1)^n * Sum[(-1)^(Mod[k+1, 2] * j) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 25 2021 *)
CROSSREFS
Columns 1-2 give A122803, A108520.
Sequence in context: A110162 A350228 A199087 * A087704 A165092 A306915
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Mar 16 2019
STATUS
approved