A071109 -id:A071109 - cp's OEIS frontend

A261566 Expansion of Product_{k>=1} (1/(1 + 2*x^k))^k.

1, -2, 0, -6, 16, -18, 48, -94, 208, -426, 752, -1646, 3360, -6578, 13056, -26358, 53456, -105890, 211392, -424366, 850544, -1699290, 3393136, -6795646, 13601184, -27188130, 54358000, -108752870, 217552976, -435033618, 869999584, -1740145118, 3480497584

Offset: 0

Vaclav Kotesovec, Aug 24 2015

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Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A071109, A255528, A261567.

nmax = 40; CoefficientList[Series[Product[(1/(1 + 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*2^k/k*x^k/(1-x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1 - 1/(-2)^j)^(j+1) = 0.81033497534928929188778847125052151513524786804782471307090750707405...

A292133 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, -1, 0, 1, -4, 6, -6, 1, 0, 1, -5, 12, -21, 14, -1, 0, 1, -6, 20, -52, 69, -26, 1, 0, 1, -7, 30, -105, 220, -201, 50, -1, 0, 1, -8, 42, -186, 545, -868, 591, -102, 2, 0, 1, -9, 56, -301, 1146, -2705, 3436, -1785, 214, -2, 0

Offset: 0

Seiichi Manyama, Sep 09 2017

			Square array begins:
   1,  1,  1,   1,   1, ...
   0, -1, -2,  -3,  -4, ...
   0,  0,  2,   6,  12, ...
   0, -1, -6, -21, -52, ...
   0,  1, 14,  69, 220, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A081362, A071109, A261582.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292134.
Cf. A246935, A286957.

A351023 Dirichlet g.f.: Product_{k>=2} 1 / (1 + 2 * k^(-s)).

Original entry on oeis.org

1, -2, -2, 2, -2, 2, -2, -6, 2, 2, -2, -2, -2, 2, 2, 14, -2, -2, -2, -2, 2, 2, -2, 10, 2, 2, -6, -2, -2, 2, -2, -26, 2, 2, 2, 6, -2, 2, 2, 10, -2, 2, -2, -2, -2, 2, -2, -18, 2, -2, 2, -2, -2, 10, 2, 10, 2, 2, -2, 2, -2, 2, -2, 50, 2, 2, -2, -2, 2, 2, -2, -14, -2, 2, -2

Offset: 1

Ilya Gutkovskiy, Jan 29 2022

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Cf. A071109, A165872, A316441, A339717, A349921, A351024.

A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, -1, -1, -3, -1, -7, -1, -15, 31, -63, 159, -95, 671, -287, 3231, -2975, 15519, -7839, 44191, -34975, 224415, -291999, 863391, -990367, 2927775, -4902047, 12561567, -27225247, 56470687, -102640799, 152153247, -422620319, 877243551, -2278272159, 3357125791

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

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Cf. A032302, A070877, A070933, A071109, A075900, A304961, A327550, A337299, A352762.

nmax = 34; CoefficientList[Series[Product[1/(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 2^(n - k), {k, 0, n}], {n, 0, 34}]

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 2^(n-k), where p(n,k) is the number of partitions of n into k parts.

cp's OEIS Frontend

A261566 Expansion of Product_{k>=1} (1/(1 + 2*x^k))^k.

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A292133 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + k*x^j).

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Examples

Links

Crossrefs

A351023 Dirichlet g.f.: Product_{k>=2} 1 / (1 + 2 * k^(-s)).

Views

Author

Keywords

Crossrefs

A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula