A290054 -id:A290054 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Jan 24 2018

nonn

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

Index entries for sequences related to sums of cubes

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

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

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

A291700 Number of ways of writing n as a sum of n nonnegative cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 9, 73, 361, 1321, 3961, 10297, 24025, 51481, 103081, 196521, 368425, 720937, 1589161, 4069801, 11511721, 33341353, 94142313, 253860201, 650564201, 1588228228, 3716917597, 8418378043, 18699454621, 41451042556, 93508305513, 218218347865, 530189399785

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for sequences related to sums of cubes

Main diagonal of A290054.

Cf. A066535, A106337, A287617.

Table[SeriesCoefficient[Sum[x^k^3, {k, 0, n}]^n, {x, 0, n}], {n, 0, 34}]

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

A363779 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^3))^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0, 1, -8, 28, -56, 70, -56, 28, -8, 0, 0, 1, -9, 36, -84, 126, -126, 84, -36, 7, 1, 0, 1, -10, 45, -120, 210, -252, 210, -120, 42, -4, -2, 0

Offset: 0

Seiichi Manyama, Jun 21 2023

			Square array begins:
  1,  1,  1,   1,   1,    1,    1, ...
  0, -1, -2,  -3,  -4,   -5,   -6, ...
  0,  1,  3,   6,  10,   15,   21, ...
  0, -1, -4, -10, -20,  -35,  -56, ...
  0,  1,  5,  15,  35,   70,  126, ...
  0, -1, -6, -21, -56, -126, -252, ...
  0,  1,  7,  28,  84,  210,  462, ...

Columns k=0..3 give A000007, A323633, A363776, A363777.
Main diagonal gives A363781.
Cf. A290054, A363778, A363783.

T(0,k) = 1; T(n,k) = -(k/n) * Sum_{j=1..n} A363783(j) * T(n-j,k).

A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^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, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 0, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 0, 1, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 3, 2, 0, 0, 1, 12, 55, 120, 135, 112, 112, 60, 15, 12, 3, 2, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 31 2017

A(n,k) is the number of ways of writing n as a sum of k tetrahedral (or triangular pyramidal) numbers (A000292).

			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,  1,  2,  3,   5,  10,  ...
0,  0,  2,  6,  12,  21,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A104246, A286180, A290054, A290430.
Cf. A000007 (column 0), A023533 (column 1), A282172 (column 5).
Main diagonal gives A303170.
Similar to, but different from, A045847.

Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (i + 2)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

A302019 Expansion of 1/(1 - x*Sum_{k>=0} x^(k^3)).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 56, 91, 149, 243, 397, 648, 1058, 1727, 2819, 4602, 7512, 12263, 20018, 32678, 53344, 87080, 142151, 232050, 378803, 618366, 1009433, 1647819, 2689933, 4391101, 7168122, 11701387, 19101580, 31181804, 50901806, 83093134, 135642908, 221426218, 361460624

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

nonn

N. J. A. Sloane, Transforms
Index entries for sequences related to sums of cubes

Antidiagonal sums of A290054.

Cf. A010057, A302018.

nmax = 41; CoefficientList[Series[1/(1 - x Sum[x^k^3, {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Sum_{k>=0} x^(k^3)).

a(0) = 1; a(n) = Sum_{k=1..n} A010057(k-1)*a(n-k).

A303169 a(n) = [x^(n^3)] (1/(1 - x))*(Sum_{k>=0} x^(k^3))^n.

Original entry on oeis.org

1, 2, 6, 30, 241, 2093, 23059, 276056, 3657901, 51751598, 792918670, 13031054778, 228632547574, 4247832219975, 83138970732860, 1710953260292025, 36844216654753387, 827664913984323748, 19363023028132371129, 470436686367280495474, 11843579175327033093769

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

Number of nonnegative solutions to (x_1)^3 + (x_2)^3 + ... + (x_n)^3 <= n^3.

Index entries for sequences related to sums of cubes

Main diagonal of A303484.

Cf. A000578, A010052, A224214, A224215, A290054, A298671, A302863.

Table[SeriesCoefficient[1/(1 - x) Sum[x^k^3, {k, 0, n}]^n, {x, 0, n^3}], {n, 0, 20}]

A303484 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^3)] (1/(1 - x))*(Sum_{j>=0} x^(j^3))^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 20, 30, 18, 6, 1, 1, 7, 37, 84, 66, 26, 7, 1, 1, 8, 70, 237, 241, 115, 37, 8, 1, 1, 9, 135, 662, 853, 500, 200, 50, 9, 1, 1, 10, 264, 1780, 2847, 2093, 1012, 302, 63, 10, 1, 1, 11, 520, 4536, 9033, 8451, 4914, 1769, 441, 80, 11, 1

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

A(n,k) is the number of nonnegative solutions to (x_1)^3 + (x_2)^3 + ... + (x_k)^3 <= n^3.

			Square array begins:
1,  1,   1,    1,    1,     1,  ...
1,  2,   3,    4,    5,     6,  ...
1,  3,   6,   11,   20,    37,  ...
1,  4,  11,   30,   84,   237,  ...
1,  5,  18,   66,  241,   853,  ...
1,  6,  26,  115,  500,  2093,  ...

Index entries for sequences related to sums of cubes

Columns k=0..4 give A000012, A000027, A224214, A224215.
Main diagonal gives A303169.
Cf. A290054, A302998.

Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^3, {i, 0, n}]^k, {x, 0, n^3}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

cp's OEIS Frontend

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

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A291700 Number of ways of writing n as a sum of n nonnegative cubes.

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A363779 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^3))^k.

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

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A302019 Expansion of 1/(1 - x*Sum_{k>=0} x^(k^3)).

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A303169 a(n) = [x^(n^3)] (1/(1 - x))*(Sum_{k>=0} x^(k^3))^n.

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A303484 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^3)] (1/(1 - x))*(Sum_{j>=0} x^(j^3))^k.

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Mathematica

A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.