A292128 -id:A292128 - cp's OEIS frontend

A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

1, -1, -2, -7, -11, -43, -65, -259, -146, -1798, 826, -8116, 17593, -35089, 301903, -308464, 3582403, 157367, 28816009, 9388694, 329375419, -61352008, 2991009094, 509592773, 23675224255, 1207374806, 229200996508, -129896994130, 2090952547882, -816324790165, 14079091274800

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A032308, A242587, A261582, A292128, A300579, A344062, A352402, A352786.

nmax = 30; CoefficientList[Series[Product[1/(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 3^(n - k), {k, 0, n}], {n, 0, 30}]

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 3^(n-k), where p(n,k) is the number of partitions of n into k parts.

A292131 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -2, 0, 0, 1, -4, -3, 2, 0, 0, 1, -5, -4, 6, 2, 1, 0, 1, -6, -5, 12, 6, 6, 0, 0, 1, -7, -6, 20, 12, 15, -2, 1, 0, 1, -8, -7, 30, 20, 28, -12, 2, 0, 0, 1, -9, -8, 42, 30, 45, -36, -3, -6, 0, 0, 1, -10, -9, 56, 42, 66, -80

Offset: 0

Seiichi Manyama, Sep 09 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -2, -3, -4, ...
   0,  0,  2,  6, 12, ...
   0,  0,  2,  6, 12, ...

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

Columns k=0..3 give A000007, A010815, A070877, A292128.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292132.
Cf. A246935, A286957.

A292130 Triangle read by rows: T(n,k) = (-3) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, -3, 0, -3, 0, -3, 9, 0, -3, 9, 0, -3, 18, 0, -3, 18, -27, 0, -3, 27, -27, 0, -3, 27, -54, 0, -3, 36, -81, 0, -3, 36, -108, 81, 0, -3, 45, -135, 81, 0, -3, 45, -189, 162, 0, -3, 54, -216, 243, 0, -3, 54, -270, 405, 0, -3, 63, -324, 486, -243, 0, -3, 63, -378

Offset: 0

Seiichi Manyama, Sep 09 2017

			First few rows are:
  1;
  0, -3;
  0, -3;
  0, -3,  9;
  0, -3,  9;
  0, -3, 18;
  0, -3, 18,  -27;
  0, -3, 27,  -27;
  0, -3, 27,  -54;
  0, -3, 36,  -81;
  0, -3, 36, -108, 81.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A292128.
Columns 0-1 give A000007, (-1)*A010701.
Cf. A291971.

A349926 Dirichlet g.f.: Product_{k>=2} (1 - 3 * k^(-s)).

Original entry on oeis.org

1, -3, -3, -3, -3, 6, -3, 6, -3, 6, -3, 15, -3, 6, 6, 6, -3, 15, -3, 15, 6, 6, -3, -3, -3, 6, 6, 15, -3, -3, -3, 15, 6, 6, 6, -3, -3, 6, 6, -3, -3, -3, -3, 15, 15, 6, -3, -21, -3, 15, 6, 15, -3, -3, 6, -3, 6, 6, -3, -39, -3, 6, 15, -12, 6, -3, -3, 15, 6, -3

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

sign

Cf. A114592, A292128, A339702, A349925.

A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

Original entry on oeis.org

1, -1, -3, -6, -18, -27, -108, -81, -486, 0, -1458, 8748, -6561, 118098, 118098, 1003833, 1417176, 11691702, 9565938, 105225318, 114791256, 746143164, 1076168025, 7231849128, 2324522934, 58113073350, 45328197213, 334731302496, 146444944842, 3263630199336, -3012581722464

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A003056, A032308, A242587, A261582, A292128, A300579, A337299, A344062, A352762.

nmax = 30; CoefficientList[Series[Product[(1 - 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 30}]

a(n) = Sum_{k=0..A003056(n)} (-1)^k * q(n,k) * 3^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

cp's OEIS Frontend

A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

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A292131 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - k*x^j).

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A292130 Triangle read by rows: T(n,k) = (-3) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Views

Author

Keywords

Examples

Links

Crossrefs

A349926 Dirichlet g.f.: Product_{k>=2} (1 - 3 * k^(-s)).

Views

Author

Keywords

Crossrefs

A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula