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 1-6 of 6 results.

A298672 Number of ordered ways of writing n^3 as a sum of n positive cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 20, 0, 1121, 72828, 872640, 9037710, 118590450, 1743739426, 24407782672, 424735169040, 7802802463460, 135385454550288, 2823521345232834, 59332856029292241, 1238888844244575904, 28893281420537822022, 684650546073054870188, 16342742577592266281996
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2018

Keywords

Examples

			a(6) = 20 because we have [64, 64, 64, 8, 8, 8], [64, 64, 8, 64, 8, 8], [64, 64, 8, 8, 64, 8], [64, 64, 8, 8, 8, 64], [64, 8, 64, 64, 8, 8], [64, 8, 64, 8, 64, 8], [64, 8, 64, 8, 8, 64], [64, 8, 8, 64, 64, 8], [64, 8, 8, 64, 8, 64], [64, 8, 8, 8, 64, 64], [8, 64, 64, 64, 8, 8], [8, 64, 64, 8, 64, 8], [8, 64, 64, 8, 8, 64], [8, 64, 8, 64, 64, 8], [8, 64, 8, 64, 8, 64], [8, 64, 8, 8, 64, 64], [8, 8, 64, 64, 64, 8], [8, 8, 64, 64, 8, 64], [8, 8, 64, 8, 64, 64] and [8, 8, 8, 64, 64, 64].

Links

Crossrefs

Cf. A000578, A030272, A232173, A259792, A290247, A291700, A298329, A298330, A298641, A298671.

Programs

  • Mathematica
    Join[{1}, Table[SeriesCoefficient[Sum[x^k^3, {k, 1, n}]^n, {x, 0, n^3}], {n, 1, 23}]]

Formula

a(n) = [x^(n^3)] (Sum_{k>=1} x^(k^3))^n.

A298671 Number of ordered ways of writing n^3 as a sum of n nonnegative cubes.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 146, 4207, 26329, 257721, 3556495, 42685181, 631230381, 9409600499, 142557084957, 2781352245050, 52598395446786, 950288577530017, 20768368026768594, 448759012546543804, 9652848877533217174, 235179507693424886403, 5756272592837812726164, 140920987987840184113287
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2018

Keywords

Examples

			a(3) = 3 because we have [27, 0, 0], [0, 27, 0] and [0, 0, 27].

Links

Crossrefs

Cf. A000578, A030272, A232173, A259792, A290054, A290247, A291700, A298329, A298330, A298641, A298672.

Programs

  • Mathematica
    Table[SeriesCoefficient[Sum[x^k^3, {k, 0, n}]^n, {x, 0, n^3}], {n, 0, 23}]

Formula

a(n) = [x^(n^3)] (Sum_{k>=0} x^(k^3))^n.

A290054 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j^3))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 0, 0, 1, 6, 10, 4, 0, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 1, 0, 1, 9, 28, 35, 15, 1, 0, 0, 2, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 3, 2, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 4, 6, 0, 0, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 19 2017

Keywords

Comments

A(n,k) is the number of ways of writing n as a sum of k nonnegative cubes.

Examples

			Square array begins:
1,  1,  1,  1,  1,   1,  ...
0,  1,  2,  3,  4,   5,  ...
0,  0,  1,  3,  6,  10,  ...
0,  0,  0,  1,  4,  10,  ...
0,  0,  0,  0,  1,   5,  ...
0,  0,  0,  0,  0,   1,  ...

Links

Crossrefs

Columns k=0-9 give: A000007, A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Main diagonal gives A291700.
Antidiagonal sums give A302019.
Cf. A045847, A122141, A286815.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Sum[x^i^3, {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

Formula

G.f. of column k: (Sum_{j>=0} x^(j^3))^k.

A298936 Number of ordered ways of writing n^2 as a sum of n nonnegative cubes.

Original entry on oeis.org

1, 1, 0, 6, 6, 20, 120, 7, 1689, 6636, 36540, 64020, 963996, 2894892, 19555965, 176079995, 955611188, 6684303780, 42462792168, 292378003753, 1886275214112, 13384059605364, 87399249887334, 624073002367892, 5080120229014734, 37587589611771480
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(3) = 6 because we have [8, 1, 0], [8, 0, 1], [1, 8, 0], [1, 0, 8], [0, 8, 1] and [0, 1, 8].

Links

Crossrefs

Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298937.

Programs

  • Maple
    f:= n -> coeff(add(x^(k^3),k=0..floor(n^(2/3)))^n,x,n^2):
map(f, [$0..30]); # Robert Israel, Jan 29 2018
  • Mathematica
    Table[SeriesCoefficient[Sum[x^k^3, {k, 0, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 0, 25}]

Formula

a(n) = [x^(n^2)] (Sum_{k>=0} x^(k^3))^n.

A298937 Number of ordered ways of writing n^2 as a sum of n positive cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 9240, 34650, 1716, 48477, 551915, 6726720, 89973520, 102639744, 1824625081, 9915389400, 30143458884, 278196062760, 1995766236541, 6611689457736, 64547920386450, 236756174748626, 2315743488707806
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(7) = 7 because we have [8, 8, 8, 8, 8, 8, 1], [8, 8, 8, 8, 8, 1, 8], [8, 8, 8, 8, 1, 8, 8], [8, 8, 8, 1, 8, 8, 8], [8, 8, 1, 8, 8, 8, 8], [8, 1, 8, 8, 8, 8, 8] and [1, 8, 8, 8, 8, 8, 8].

Links

Crossrefs

Cf. A000290, A000578, A023358, A030272, A218495, A259792, A291700, A298671, A298672, A298934, A298936.

Programs

  • Mathematica
    Join[{1}, Table[SeriesCoefficient[Sum[x^k^3, {k, 1, Floor[n^(2/3) + 1]}]^n, {x, 0, n^2}], {n, 1, 27}]]

Formula

a(n) = [x^(n^2)] (Sum_{k>=1} x^(k^3))^n.

A363781 a(n) = [x^n] 1/(Sum_{k>=0} x^(k^3))^n.

Original entry on oeis.org

1, -1, 3, -10, 35, -126, 462, -1716, 6427, -24220, 91718, -348712, 1330238, -5088916, 19515660, -75000600, 288772795, -1113688667, 4301387580, -16635049690, 64410064110, -249659926890, 968651123520, -3761601535620, 14619516646150, -56861702382576
Offset: 0

Views

Author

Seiichi Manyama, Jun 21 2023

Keywords

Crossrefs

Main diagonal of A363779.
Cf. A291700.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.