A306846
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).
Original entry on oeis.org
1, 1, 2, 1, 1, 4, 1, 1, 2, 8, 1, 1, 1, 4, 16, 1, 1, 1, 2, 8, 32, 1, 1, 1, 1, 5, 16, 64, 1, 1, 1, 1, 2, 11, 32, 128, 1, 1, 1, 1, 1, 6, 22, 64, 256, 1, 1, 1, 1, 1, 2, 16, 43, 128, 512, 1, 1, 1, 1, 1, 1, 7, 36, 85, 256, 1024, 1, 1, 1, 1, 1, 1, 2, 22, 72, 170, 512, 2048
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 1, 1, 1, 1, 1, 1, 1, 1, ...
4, 2, 1, 1, 1, 1, 1, 1, 1, ...
8, 4, 2, 1, 1, 1, 1, 1, 1, ...
16, 8, 5, 2, 1, 1, 1, 1, 1, ...
32, 16, 11, 6, 2, 1, 1, 1, 1, ...
64, 32, 22, 16, 7, 2, 1, 1, 1, ...
128, 64, 43, 36, 22, 8, 2, 1, 1, ...
256, 128, 85, 72, 57, 29, 9, 2, 1, ...
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T[n_, k_] := Sum[Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)
A306939
Expansion of 1/((1 - x)^9 - x^9).
Original entry on oeis.org
1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24311, 43776, 75753, 127110, 209475, 346104, 591261, 1081575, 2163150, 4686826, 10656387, 24582663, 56191734, 125640180, 273241161, 577147212, 1184959314, 2369918628, 4631710931, 8881943832, 16798969548, 31537530456
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..3000
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,2).
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CoefficientList[Series[1/((1 - x)^9 - x^9), {x, 0, 30}], x] (* Amiram Eldar, May 25 2021 *)
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{a(n) = sum(k=0, n\9, binomial(n+8, 9*k+8))}
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N=66; x='x+O('x^N); Vec(1/((1-x)^9-x^9))
A307045
a(n) = Sum_{k=0..floor(n/9)} (-1)^k*binomial(n,9*k).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -9, -54, -219, -714, -2001, -5004, -11439, -24309, -48618, -92358, -167769, -292599, -490104, -783540, -1172907, -1562274, -1562274, 0, 6216183, 24581880, 72182016, 186061536, 443185425, 997483653, 2146130559, 4443424371, 8886848742
Offset: 0
- Seiichi Manyama, Table of n, a(n) for n = 0..3000
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9).
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a[n_] := Sum[(-1)^k * Binomial[n,9*k], {k,0,Floor[n/9]}]; Array[a, 37, 0] (* Amiram Eldar, May 25 2021 *)
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{a(n) = sum(k=0, n\9, (-1)^k*binomial(n, 9*k))}
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N=66; x='x+O('x^N); Vec((1-x)^8/((1-x)^9+x^9))
Showing 1-3 of 3 results.