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.

A285047 Expansion of Product_{k>=1} 1/(1-x^(k^2))^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 6, 9, 9, 9, 13, 19, 19, 19, 28, 37, 43, 43, 57, 69, 81, 81, 100, 132, 150, 160, 184, 236, 260, 280, 319, 391, 460, 490, 565, 657, 771, 811, 922, 1084, 1243, 1363, 1510, 1781, 1985, 2185, 2388, 2775, 3159, 3439, 3832, 4335, 4963, 5323
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A000219, A001156, A023871, A266941, A281790.

Programs

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

Formula

log(a(n)) ~ Pi*sqrt(n/3).

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

A291655 Expansion of Product_{k>=1} 1/(1-x^(k^2))^(k^2).

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 5, 5, 15, 24, 24, 24, 44, 80, 80, 80, 131, 221, 266, 266, 386, 566, 746, 746, 990, 1474, 1924, 2089, 2529, 3709, 4609, 5269, 6130, 8576, 11096, 12746, 14937, 19397, 25697, 28997, 34111, 43135, 56365, 65905, 76219, 95770, 120070, 144370, 163661
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 28 2017

Keywords

Links

Crossrefs

Cf. A001156, A033461, A281790, A285047, A291649, A291696, A291720.

Programs

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

Formula

log(a(n)) ~ 5 * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5) / (2^(6/5) * 3^(3/5)).

A285242 Expansion of Product_{k>=1} (1 + k*x^(k^2))^k.

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 0, 4, 13, 9, 0, 0, 36, 36, 0, 16, 52, 63, 27, 64, 64, 108, 108, 64, 233, 277, 135, 27, 676, 676, 108, 204, 772, 1333, 765, 528, 420, 2628, 2628, 528, 1792, 3892, 3735, 1251, 5524, 5380, 4428, 4684, 6657, 12843, 10870, 6703, 3767, 28232
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A022629, A266891, A281790, A285240, A285243, A285244.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + k*x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k, j]*k^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]

A291667 Expansion of Product_{k>=1} ((1 + x^(k^2)) / (1 - x^(k^2)))^k.

Original entry on oeis.org

1, 2, 2, 2, 6, 10, 10, 10, 18, 32, 38, 38, 50, 86, 110, 110, 134, 206, 272, 290, 342, 466, 610, 682, 770, 1012, 1310, 1492, 1654, 2130, 2698, 3066, 3410, 4210, 5310, 6106, 6812, 8078, 10118, 11750, 13006, 15198, 18654, 21810, 24178, 28092, 33682, 39330, 43866
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 29 2017

Keywords

Comments

Convolution of A285047 and A281790.

Links

Crossrefs

Cf. A001156, A033461, A285047, A281790, A291666.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[((1+x^(k^2))/(1-x^(k^2)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

log(a(n)) ~ Pi*sqrt(n/2).

A301518 a(n) = [x^n] Product_{k>=1} (1 + x^(k^2))^n.

Original entry on oeis.org

1, 1, 1, 1, 5, 26, 91, 246, 589, 1468, 4226, 13311, 41471, 122864, 351184, 1001876, 2920957, 8698612, 26070130, 77707056, 229959130, 679050870, 2011457295, 5986185690, 17866178695, 53343031301, 159149943668, 474683353849, 1416730630936, 4233405443596
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 23 2018

Keywords

Links

Crossrefs

Cf. A033461, A281790, A291649, A300974.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + x^(k^2))^n, {k, 1, n}], {x, 0, n}], {n, 0, 30}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = 3.04074590736461391963643911... and c = 0.2268848664201836146769277...
Showing 1-6 of 6 results.

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