cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A283272 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is expansion of Product_{j>=1} (1-x^j)^(j^k) in power of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -2, 0, 1, -1, -4, -1, 0, 1, -1, -8, -5, 0, 1, 1, -1, -16, -19, -1, 4, 0, 1, -1, -32, -65, -9, 21, 4, 1, 1, -1, -64, -211, -55, 127, 49, 7, 0, 1, -1, -128, -665, -285, 807, 500, 81, 3, 0, 1, -1, -256, -2059, -1351, 5179, 4809, 1038, 45
Offset: 0

Views

Author

Seiichi Manyama, Mar 04 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,   1,   1,    1, ...
  -1, -1, -1,  -1,  -1,   -1, ...
  -1, -2, -4,  -8, -16,  -32, ...
   0, -1, -5, -19, -65, -211, ...
   0,  0, -1,  -9, -55, -285, ...
   1,  4, 21, 127, 807, 5179, ...

Links

Crossrefs

Columns k=0-9 give A010815, A073592, A283263, A283264, A283271, A283336, A283337, A283338, A283339, A283340.
Row k=5 gives A281581.
Main diagonal gives A283333.
Cf. A144048.

Formula

G.f. of column k: Product_{j>=1} (1-x^j)^(j^k).

A300457 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).

Original entry on oeis.org

1, -1, -3, -1, 25, 624, 9871, 170470, 3027249, 55077245, 979330606, 15079702923, 94670678245, -7958168036625, -626145997536240, -34564907982551791, -1733699815491494303, -84294315853736719077, -4067859614343931897505, -196552300464314521511610, -9519733465269825759734169
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of Product_{k>=1} (1 - x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),  -1,    0,   0,     1,  ...
n = 2:  1,  -2,  (-3),   0,   2,    12,  ...
n = 3:  1,  -3,   -6,  (-1),  9,    63,  ...
n = 4:  1,  -4,  -10,   -4, (25),  224,  ...
n = 5:  1,  -5,  -15,  -10,  55,  (624), ...

Crossrefs

Cf. A010815, A008705, A252654, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300458.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

Original entry on oeis.org

1, -1, -1, -10, 11, 374, 9792, 183847, 3469427, 65038049, 1195396233, 19667738452, 189089161562, -6219720781782, -606316892131934, -35104997710496175, -1795953382595105853, -88223902016631657740, -4283800987347611165184, -207864171877269042498096, -10102590396625592962089500
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of Product_{k>=1} 1/(1 + x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),   0,   -1,   1,    -1,  ...
n = 2:  1,  -2,  (-1),  -4,   3,    -2,  ...
n = 3:  1,  -3,   -3, (-10),  6,    15,  ...
n = 4:  1,  -4,   -6,  -20, (11),  104,  ...
n = 5:  1,  -5,  -10,  -35,  20,  (374), ...

Crossrefs

Cf. A081362, A252654, A255526, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300457.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 + x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

A321260 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(sigma_n(k)-k^n).

Original entry on oeis.org

1, 0, 1, 1, 18, 2, 861, 132, 106024, 40910, 72980055, 6838271, 228282942581, 27620223647, 2050169324675668, 352809815149813, 87174966874755673105, 6798293425492905407, 18318448554980083512011863, 1187839217207171380193247, 11258918803635775614062752424535
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Crossrefs

Cf. A283333, A318783, A318784, A319647, A321258, A321261.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k)^(DivisorSigma[n, k] - k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^(2 k)/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^(2*k)/(k*(1 - x^k))).
Showing 1-4 of 4 results.

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