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, ...
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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 *)
A307394
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, 3, 1, 4, 6, 1, 4, 9, 10, 1, 4, 10, 14, 15, 1, 4, 10, 19, 15, 21, 1, 4, 10, 20, 28, 8, 28, 1, 4, 10, 20, 34, 28, -7, 36, 1, 4, 10, 20, 35, 48, 1, -22, 45, 1, 4, 10, 20, 35, 55, 48, -80, -21, 55, 1, 4, 10, 20, 35, 56, 75, 0, -242, 12, 66, 1, 4, 10, 20, 35, 56, 83, 75, -164, -485, 77, 78
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3, 4, 4, 4, 4, 4, 4, 4, 4, ...
6, 9, 10, 10, 10, 10, 10, 10, 10, ...
10, 14, 19, 20, 20, 20, 20, 20, 20, ...
15, 15, 28, 34, 35, 35, 35, 35, 35, ...
21, 8, 28, 48, 55, 56, 56, 56, 56, ...
28, -7, 1, 48, 75, 83, 84, 84, 84, ...
36, -22, -80, 0, 75, 110, 119, 120, 120, ...
45, -21, -242, -164, 0, 110, 154, 164, 165, ...
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T[n_, k_] := Sum[(-1)^j * Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)
A307078
Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).
Original entry on oeis.org
1, 1, 3, 1, 2, 7, 1, 2, 4, 15, 1, 2, 3, 8, 31, 1, 2, 3, 5, 16, 63, 1, 2, 3, 4, 10, 32, 127, 1, 2, 3, 4, 6, 21, 64, 255, 1, 2, 3, 4, 5, 12, 43, 128, 511, 1, 2, 3, 4, 5, 7, 28, 86, 256, 1023, 1, 2, 3, 4, 5, 6, 14, 64, 171, 512, 2047, 1, 2, 3, 4, 5, 6, 8, 36, 136, 341, 1024, 4095
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3, 2, 2, 2, 2, 2, 2, 2, 2, ...
7, 4, 3, 3, 3, 3, 3, 3, 3, ...
15, 8, 5, 4, 4, 4, 4, 4, 4, ...
31, 16, 10, 6, 5, 5, 5, 5, 5, ...
63, 32, 21, 12, 7, 6, 6, 6, 6, ...
127, 64, 43, 28, 14, 8, 7, 7, 7, ...
255, 128, 86, 64, 36, 16, 9, 8, 8, ...
511, 256, 171, 136, 93, 45, 18, 10, 9, ...
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T[n_, k_] := Sum[Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)
A307089
Expansion of (1 - x)^4/((1 - x)^6 + x^6).
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 6, 0, -27, -110, -319, -780, -1702, -3404, -6315, -10864, -17051, -23238, -23238, 0, 87021, 325358, 890077, 2107560, 4542526, 9085052, 16950573, 29354524, 46296905, 63239286, 63239286, 0, -236031147, -880918070, -2406788599, -5694626340
Offset: 0
-
a[n_] := Sum[(-1)^k * Binomial[n+1, 6*k+1], {k, 0, Floor[n/6]}]; Array[a, 36, 0] (* Amiram Eldar, May 14 2021 *)
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{a(n) = sum(k=0, n\6, (-1)^k*binomial(n+1,6*k+1))}
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N=66; x='x+O('x^N); Vec((1-x)^4/((1-x)^6+x^6))
Showing 1-4 of 4 results.