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 1-10 of 13 results. Next

A298330 Number of ordered ways of writing n^2 as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 0, 3, 1, 5, 141, 742, 6120, 43888, 300232, 3074478, 28901797, 290411147, 3175037698, 34951274416, 399750066121, 4814421349467, 59532792202344, 768079420764884, 10247011240209066, 140144002390928732, 1978092111496441512, 28633995987157024399
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 17 2018

Keywords

Examples

			a(3) = 3 because we have [4, 4, 1], [4, 1, 4] and [1, 4, 4].

Links

Crossrefs

Cf. A037444, A066535, A232173, A287617, A298329.

Programs

  • Maple
    G:= (JacobiTheta3(0,x)-1)/2:
f:= proc(n) local S; S:= series(G^n,x,n^2+1); coeff(S,x,n^2) end proc:
map(f, [$0..25]); # Robert Israel, Dec 16 2024
  • Mathematica
    Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^2}], {n, 0, 23}]

Formula

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

A298329 Number of ordered ways of writing n^2 as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 2, 6, 5, 90, 582, 4081, 45678, 378049, 3844532, 39039539, 395170118, 4589810849, 53154371025, 660113986997, 8584476248237, 113555197832758, 1572878837435750, 22259911738401660, 324143769099772448, 4869443438412466557, 74837370448784241452, 1182177603062005007658
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 17 2018

Keywords

Examples

			a(3) = 6 because we have [9, 0, 0], [4, 4, 1], [4, 1, 4], [1, 4, 4], [0, 9, 0] and [0, 0, 9].

Links

Crossrefs

[x^(n^b)] (Sum_{k>=0} x^(k^b))^n: A088218 (b=1), this sequence (b=2), A298671 (b=3).
Cf. A037444, A066535, A232173, A287617, A298330, A300446.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, 1/t!, (s->
     `if`(s*t n!*b(n^2, n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 28 2018
  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^2}], {n, 0, 23}]
  • PARI
    {a(n) = polcoeff((sum(k=0, n, x^(k^2)+x*O(x^(n^2))))^n, n^2)} \\ Seiichi Manyama, Oct 28 2018

Formula

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

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.

A300446 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 8, 12, 12, 17, 23, 27, 32, 41, 52, 61, 77, 91, 110, 134, 159, 188, 228, 271, 314, 380, 444, 518, 612, 713, 832, 976, 1128, 1308, 1529, 1756, 2023, 2343, 2698, 3091, 3555, 4072, 4657, 5343, 6074, 6922, 7912, 8986, 10194, 11590, 13135, 14855
Offset: 0

Views

Author

Seiichi Manyama, May 11 2018

Keywords

Comments

Also the number of partitions of n in which each part occurs a square number (>=0) of times.

Examples

			n | Partitions of n in which each part occurs a square number (>=0) of times
--+-------------------------------------------------------------------------
1 | 1;
2 | 2;
3 | 3 = 2+1;
4 | 4 = 3+1 = 1+1+1+1;
5 | 5 = 4+1 = 3+2;
6 | 6 = 5+1 = 4+2 = 3+2+1 = 2+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 3+1+1+1+1;
8 | 8 = 7+1 = 6+2 = 5+3 = 5+2+1 = 4+3+1 = 4+1+1+1+1 = 2+2+2+2;

Links

Crossrefs

Cf. A000041, A000122, A010052, A158441, A232173, A298329, A304329, A320932, A321139.

Programs

  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&+[x^(k*j^2):j in [0..2*m]]): k in [1..2*m]]) ));  // G. C. Greubel, Oct 29 2018
  • Maple
    b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018
  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 25 2018 *)
  • PARI
    N=99; x='x+O('x^N); Vec(prod(i=1, N, sum(j=0, sqrtint(N\i), x^(i*j^2)))) \\ Seiichi Manyama, Oct 28 2018

Formula

G.f.: Product_{k>=1} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Oct 25 2018

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.

A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 4, 2, 0, 1, 8, 6, 4, 2, 0, 1, 10, 24, 30, 4, 2, 0, 1, 12, 90, 104, 6, 12, 2, 0, 1, 14, 252, 250, 24, 30, 4, 2, 0, 1, 16, 574, 876, 730, 248, 30, 4, 2, 0, 1, 18, 1136, 3542, 4092, 1210, 312, 54, 4, 2, 0, 1, 20, 2034, 12112, 18494, 7812, 2250, 456, 6, 4, 2, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2018

Keywords

Comments

A(n,k) is the number of ordered ways of writing n^2 as a sum of k squares.

Examples

			Square array begins:
  1,  1,   1,   1,    1,     1,  ...
  0,  2,   4,   6,    8,    10,  ...
  0,  2,   4,   6,   24,    90,  ...
  0,  2,   4,  30,  104,   250,  ...
  0,  2,   4,   6,   24,   730,  ...
  0,  2,  12,  30,  248,  1210,  ...

Links

Crossrefs

Columns k=0..4,7 give A000007, A040000, A046109, A016725, A267326, A361695.
Main diagonal gives A232173.
Cf. A000122, A122141, A255212, A286815.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1)+2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
A:= (n, k)-> b(n^2, k):
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 10 2023
  • Mathematica
    Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

A(n,k) = [x^(n^2)] (Sum_{j=-infinity..infinity} x^(j^2))^k.

A302861 a(n) = [x^(n^2)] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 3, 13, 123, 1281, 16875, 252673, 4031123, 70554353, 1318315075, 26107328109, 549772933959, 12147113355505, 280978137279483, 6780378828922333, 169829490474843659, 4409771551548703649, 118361723203178140163, 3277041835527134201777, 93455465161026267454527
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 14 2018

Keywords

Comments

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius n.

Links

Crossrefs

Main diagonal of A302997.
Cf. A000122, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416, A122510, A232173, A302860.

Programs

  • Mathematica
    Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n^2}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n^2}], {n, 0, 19}]

Formula

a(n) = A122510(n,n^2).

A298938 Number of ordered ways of writing n^3 as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 4, 5, 686, 13942, 455988, 13617853, 454222894, 18323165948, 802161109047, 42149084452070, 2481730049781672, 157265294178424356, 10977302934685469078, 812821237985857557677, 64539935903231450294134, 5504599828399250884049308
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(4) = 5 because we have [64, 0, 0, 0], [16, 16, 16, 16], [0, 64, 0, 0], [0, 0, 64, 0] and [0, 0, 0, 64].

Links

Crossrefs

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298936, A298939.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

Formula

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

A298939 Number of ordered ways of writing n^3 as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 1, 4, 1, 286, 7582, 202028, 6473625, 226029577, 8338249868, 391526193477, 19990594900630, 1159906506684446, 74890158861242740, 5119732406649036418, 380146984328280974281, 30198665638519565614034, 2555354508318427693497565
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2018

Keywords

Examples

			a(3) = 4 because we have [25, 1, 1], [9, 9, 9], [1, 25, 1] and [1, 1, 25].

Links

Crossrefs

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298937, A298938.

Programs

  • Mathematica
    Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

Formula

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

A331884 Number of compositions (ordered partitions) of n^2 into distinct squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 7, 1, 31, 123, 151, 121, 897, 7351, 5415, 14881, 48705, 150583, 468973, 1013163, 1432471, 1730023, 50432107, 14925241, 125269841, 74592537, 241763479, 213156871, 895153173, 7716880623, 2681163865, 3190865761, 22501985413, 116279718801
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2020

Keywords

Examples

			a(5) = 3 because we have [25], [16, 9] and [9, 16].

Links

Crossrefs

Cf. A000290, A006456, A030273, A032020, A037444, A105152, A224366, A232173, A280129, A298640, A331844.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n^2, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 30 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n^2, n, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

Formula

a(n) = A331844(A000290(n)).

Extensions

a(24)-a(34) from Alois P. Heinz, Jan 30 2020
Showing 1-10 of 13 results. Next

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

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