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 *)
A307393
Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k-x^k).
Original entry on oeis.org
1, 1, 5, 1, 4, 16, 1, 4, 11, 42, 1, 4, 10, 26, 99, 1, 4, 10, 21, 57, 219, 1, 4, 10, 20, 42, 120, 466, 1, 4, 10, 20, 36, 84, 247, 968, 1, 4, 10, 20, 35, 64, 169, 502, 1981, 1, 4, 10, 20, 35, 57, 120, 340, 1013, 4017, 1, 4, 10, 20, 35, 56, 93, 240, 682, 2036, 8100
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
5, 4, 4, 4, 4, 4, 4, 4, ...
16, 11, 10, 10, 10, 10, 10, 10, ...
42, 26, 21, 20, 20, 20, 20, 20, ...
99, 57, 42, 36, 35, 35, 35, 35, ...
219, 120, 84, 64, 57, 56, 56, 56, ...
466, 247, 169, 120, 93, 85, 84, 84, ...
968, 502, 340, 240, 165, 130, 121, 120, ...
-
T[n_, k_] := Sum[Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)
A307395
Expansion of 1/((1 - x) * ((1 - x)^3 + x^3)).
Original entry on oeis.org
1, 4, 10, 19, 28, 28, 1, -80, -242, -485, -728, -728, 1, 2188, 6562, 13123, 19684, 19684, 1, -59048, -177146, -354293, -531440, -531440, 1, 1594324, 4782970, 9565939, 14348908, 14348908, 1, -43046720, -129140162, -258280325, -387420488, -387420488, 1, 1162261468
Offset: 0
-
LinearRecurrence[{4, -6, 3}, {1, 4, 10}, 38] (* Amiram Eldar, May 13 2021 *)
-
{a(n) = sum(k=0, n\3, (-1)^k*binomial(n+3, 3*k+3))}
-
N=66; x='x+O('x^N); Vec(1/((1-x)*((1-x)^3+x^3)))
A307668
A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j*binomial(2*n,k*j+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 1, 2, 5, 10, 1, 2, 6, 14, 35, 1, 2, 6, 19, 43, 126, 1, 2, 6, 20, 62, 142, 462, 1, 2, 6, 20, 69, 207, 494, 1716, 1, 2, 6, 20, 70, 242, 705, 1780, 6435, 1, 2, 6, 20, 70, 251, 858, 2445, 6563, 24310, 1, 2, 6, 20, 70, 252, 912, 3068, 8622, 24566, 92378
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, 2, ...
3, 5, 6, 6, 6, 6, 6, ...
10, 14, 19, 20, 20, 20, 20, ...
35, 43, 62, 69, 70, 70, 70, ...
126, 142, 207, 242, 251, 252, 252, ...
462, 494, 705, 858, 912, 923, 924, ...
1716, 1780, 2445, 3068, 3341, 3418, 3431, ...
6435, 6563, 8622, 11051, 12310, 12750, 12854, ...
-
T[n_, k_] := Sum[(-1)^j*Binomial[2*n, k*j + n], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 13 2021*)
Showing 1-4 of 4 results.