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 1-5 of 5 results.

A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 0, 6, 15, 9, 0, 4, 40, 36, 0, 17, 71, 90, 36, 64, 100, 180, 144, 96, 274, 394, 300, 148, 740, 820, 480, 472, 1150, 1851, 1341, 1146, 1318, 3880, 3540, 1704, 3017, 6455, 7134, 3780, 7822, 9574, 12180, 10304, 12057, 19750, 22485, 20558, 15910, 43076, 43236, 31104, 33742, 66895
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 28 2017

Keywords

Comments

Number of partitions of n into distinct squares, where k^2 different parts of size k^2 are available (1a, 4a, 4b, 4c, 4d, ...).

Examples

			a(8) = 6 because we have [4a, 4b], [4a, 4c], [4a, 4d], [4b, 4c], [4b, 4d] and [4c, 4d].

Links

Crossrefs

Cf. A000290, A026007, A027998, A033461, A262736, A281790, A284896, A291655, A291692.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^k^2)^k^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k^2, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 28 2017 *)

Formula

G.f.: Product_{k>=1} (1 + x^A000290(k))^A000290(k).
a(n) ~ exp(5 * 2^(-9/5) * 3^(-3/5) * (9-4*sqrt(2))^(1/5) * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5)) * 3^(1/5) * (2*sqrt(2)-1)^(1/5) * Zeta(5/2)^(1/5) / (2^(9/10) * sqrt(5) * Pi^(2/5) * n^(7/10)). - Vaclav Kotesovec, Aug 29 2017

A291720 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^(k^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 9, 9, 9, 9, 45, 45, 45, 45, 45, 45, 45, 45, 165, 165, 165, 192, 192, 192, 192, 192, 522, 522, 522, 738, 738, 738, 738, 738, 1530, 1530, 1530, 2502, 2502, 2502, 2502, 2502, 4218, 4218, 4218, 7458, 7458, 7458, 7836, 7836
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 30 2017

Keywords

Links

Crossrefs

Cf. A000219, A003108, A291655, A291692, A291721.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1/(1 - x^(k^3))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

log(a(n)) ~ 7 * (Gamma(1/3) * Zeta(7/3))^(3/7) * n^(4/7) / (2^(8/7) * 3^(9/7)).

A291666 Expansion of Product_{k>=1} ((1 + x^(k^2)) / (1 - x^(k^2)))^(k^2).

Original entry on oeis.org

1, 2, 2, 2, 10, 18, 18, 18, 50, 100, 118, 118, 206, 438, 582, 582, 806, 1606, 2344, 2506, 3122, 5322, 8202, 9498, 11130, 16844, 26110, 32272, 37018, 52274, 78018, 100098, 115986, 155026, 223190, 291674, 345132, 439518, 618734, 811790, 972846, 1204190, 1653726
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 29 2017

Keywords

Comments

Convolution of A291649 and A291655.

Links

Crossrefs

Cf. A001156, A033461, A103265, A156616, A206622, A291649, A291655, A291667, A291721.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[((1+x^(k^2))/(1-x^(k^2)))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

log(a(n)) ~ 5 * Pi^(1/5) * ((8-sqrt(2)) * Zeta(5/2))^(2/5) * n^(3/5) / (4*3^(3/5)).

A291696 Expansion of Product_{k>=1} (1-x^(k^2))^(k^2).

Original entry on oeis.org

1, -1, 0, 0, -4, 4, 0, 0, 6, -15, 9, 0, -4, 40, -36, 0, -15, -39, 90, -36, 64, -28, -180, 144, -96, 206, 106, -300, 148, -540, 332, 480, -232, 610, -1029, -189, 114, -86, 1880, -1068, 24, -921, -1545, 2466, -300, 2858, -1514, -3180, 976, -4121, 5590, 1995
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 30 2017

Keywords

Links

Crossrefs

Cf. A073592, A291649, A291655.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1-x^(k^2))^(k^2), {k,1,nmax}], {x,0,nmax}], x]

A298730 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^(k*(k+1)/2).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 16, 16, 16, 44, 54, 54, 126, 156, 156, 315, 435, 435, 780, 1060, 1115, 1856, 2576, 2741, 4214, 5804, 6464, 9446, 12924, 14464, 20324, 27818, 31778, 43166, 58232, 66977, 89396, 120000, 139255, 181274, 240405, 282000, 362457, 476334, 560709, 708893, 924923, 1096773, 1372597
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2018

Keywords

Crossrefs

Cf. A000217, A000294, A007294, A291655, A298850.

Programs

  • Mathematica
    nmax = 48; CoefficientList[Series[Product[1/(1 - x^(k (k + 1)/2))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^A000217(k))^A000217(k).
Showing 1-5 of 5 results.

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