A306914
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).
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 0, 0, 1, 5, 10, 9, -4, 0, 1, 6, 15, 20, 9, -8, 0, 1, 7, 21, 35, 34, 0, -8, 0, 1, 8, 28, 56, 70, 48, -27, 0, 0, 1, 9, 36, 84, 126, 125, 48, -81, 16, 0, 1, 10, 45, 120, 210, 252, 200, 0, -162, 32, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 2, 3, 4, 5, 6, 7, 8, ...
0, 2, 6, 10, 15, 21, 28, 36, ...
0, 0, 9, 20, 35, 56, 84, 120, ...
0, -4, 9, 34, 70, 126, 210, 330, ...
0, -8, 0, 48, 125, 252, 462, 792, ...
0, -8, -27, 48, 200, 461, 924, 1716, ...
0, 0, -81, 0, 275, 780, 1715, 3432, ...
0, 16, -162, -164, 275, 1209, 2989, 6434, ...
-
A[n_, k_] := SeriesCoefficient[1/((1-x)^k + x^k), {x, 0, n}];
Table[A[n-k+1, k], {n, 0, 11}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Mar 20 2019 *)
A306915
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).
Original entry on oeis.org
1, 1, 2, 1, 2, 4, 1, 3, 4, 8, 1, 4, 6, 8, 16, 1, 5, 10, 11, 16, 32, 1, 6, 15, 20, 21, 32, 64, 1, 7, 21, 35, 36, 42, 64, 128, 1, 8, 28, 56, 70, 64, 85, 128, 256, 1, 9, 36, 84, 126, 127, 120, 171, 256, 512, 1, 10, 45, 120, 210, 252, 220, 240, 342, 512, 1024
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
2, 2, 3, 4, 5, 6, 7, 8, ...
4, 4, 6, 10, 15, 21, 28, 36, ...
8, 8, 11, 20, 35, 56, 84, 120, ...
16, 16, 21, 36, 70, 126, 210, 330, ...
32, 32, 42, 64, 127, 252, 462, 792, ...
64, 64, 85, 120, 220, 463, 924, 1716, ...
128, 128, 171, 240, 385, 804, 1717, 3432, ...
256, 256, 342, 496, 715, 1365, 3017, 6436, ...
-
A[n_, k_] := Sum[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 *)
A307047
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).
Original entry on oeis.org
1, 1, 0, 1, -2, 0, 1, -3, 4, 0, 1, -4, 6, -8, 0, 1, -5, 10, -9, 16, 0, 1, -6, 15, -20, 9, -32, 0, 1, -7, 21, -35, 36, 0, 64, 0, 1, -8, 28, -56, 70, -64, -27, -128, 0, 1, -9, 36, -84, 126, -125, 120, 81, 256, 0, 1, -10, 45, -120, 210, -252, 200, -240, -162, -512, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, -2, -3, -4, -5, -6, -7, -8, ...
0, 4, 6, 10, 15, 21, 28, 36, ...
0, -8, -9, -20, -35, -56, -84, -120, ...
0, 16, 9, 36, 70, 126, 210, 330, ...
0, -32, 0, -64, -125, -252, -462, -792, ...
0, 64, -27, 120, 200, 463, 924, 1716, ...
0, -128, 81, -240, -275, -804, -1715, -3432, ...
0, 256, -162, 496, 275, 1365, 2989, 6436, ...
-
T[n_, k_] := (-1)^n * Sum[(-1)^(j * Mod[k, 2]) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)
Showing 1-3 of 3 results.