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.

A369520 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^k)).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 34, 52, 80, 119, 175, 251, 359, 504, 702, 965, 1320, 1785, 2401, 3200, 4245, 5589, 7324, 9535, 12364, 15944, 20478, 26175, 33338, 42279, 53438, 67283, 84454, 105642, 131764, 163826, 203149, 251185, 309799, 381079, 467666, 572520, 699342, 852314
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 25 2024

Keywords

Comments

Convolution of A001156 and A000041.
a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k.

Links

Crossrefs

Cf. A000041, A001156, A087153, A264393, A280204, A369519, A369579.

Programs

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

Formula

a(n) ~ exp(Pi*sqrt(2*n/3) + 3^(1/4)*zeta(3/2)*n^(1/4)/2^(3/4) - 3*zeta(3/2)^2/(32*Pi)) / (2^(13/4) * 3^(3/4) * sqrt(Pi) * n^(5/4)).

A280278 G.f.: Product_{k>=1} (1 + x^(k^3)) / (1 - x^k).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 26, 38, 54, 75, 103, 141, 190, 254, 337, 444, 580, 754, 973, 1250, 1597, 2030, 2568, 3237, 4061, 5076, 6322, 7847, 9705, 11968, 14711, 18033, 22043, 26873, 32677, 39642, 47972, 57924, 69787, 83904, 100667, 120547, 144072, 171876, 204677
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 30 2016

Keywords

Comments

Convolution of A279329 and A000041.

Links

Crossrefs

Cf. A000041, A264393, A279329, A280204, A369571, A369579.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1+x^(k^3))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(2*n/3) + (2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3) * n^(1/6) / (2^(1/6) * 3^(5/6) * Pi^(1/3))) / (4*sqrt(6)*n).

A369519 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^(k^3))).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 21, 26, 31, 38, 46, 54, 62, 74, 88, 103, 118, 137, 158, 180, 202, 230, 263, 298, 335, 378, 426, 476, 528, 589, 658, 732, 810, 900, 998, 1101, 1208, 1330, 1465, 1608, 1760, 1930, 2116, 2310, 2513, 2738, 2985, 3246, 3521, 3826, 4156
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 25 2024

Keywords

Comments

Convolution of A001156 and A003108.
a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k into cubes.

Links

Crossrefs

Cf. A001156, A003108.
Cf. A087153, A131799, A264393, A369520, A369579.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Product[1/(1-x^(k^2))/(1-x^(k^3)), {k, 1, nmax^(1/2)}], {x, 0, nmax}], x]

Formula

a(n) ~ zeta(3/2) * exp(3 * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3) + 2^(4/9) * Gamma(1/3) * zeta(4/3) * n^(2/9) / (3 * Pi^(1/9) * zeta(3/2)^(2/9)) - 4*2^(2/9) * Gamma(1/3)^2 * zeta(4/3)^2 * n^(1/9) / (243 * Pi^(5/9) * zeta(3/2)^(10/9)) + 16*Gamma(1/3)^3 * zeta(4/3)^3 / (6561 * Pi * zeta(3/2)^2)) / (16 * sqrt(6) * Pi^(5/2) * n^(3/2)) * (1 + (13*2^(7/9) * Gamma(1/3) * zeta(4/3) / (81 * Pi^(4/9) * zeta(3/2)^(8/9)) + 832*2^(7/9) * Gamma(1/3)^4 * zeta(4/3)^4 / (1594323 * Pi^(13/9) * zeta(3/2)^(26/9))) / n^(1/9) + (692224 * 2^(5/9) * Gamma(1/3)^8 * zeta(4/3)^8 / (2541865828329 * Pi^(26/9) * zeta(3/2)^(52/9)) - 128 * 2^(5/9) * Gamma(1/3)^5 * zeta(4/3)^5 / (4782969 * Pi^(17/9) * zeta(3/2)^(34/9)) + 65*2^(5/9) * Gamma(1/3)^2 * zeta(4/3)^2 / (2187*Pi^(8/9) * zeta(3/2)^(16/9)))/n^(2/9)).

A369576 Expansion of Product_{k>=1} 1 / ((1 - x^k) * (1 - x^(k^2)) * (1 - x^(k^3))).

Original entry on oeis.org

1, 3, 7, 14, 27, 48, 82, 134, 215, 336, 515, 773, 1145, 1670, 2406, 3423, 4824, 6730, 9310, 12768, 17385, 23499, 31559, 42111, 55876, 73726, 96784, 126418, 164375, 212772, 274277, 352125, 450365, 573891, 728765, 922305, 1163530, 1463287, 1834842, 2294128, 2860538
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

Comments

Convolution of A000041 and A001156 and A003108.
Convolution of A369520 and A003108.
a(n) is the number of triples (R(r), S(s), T(t)) where r + s + t = n, and R(k) is a partition of k, S(k) a partition of k into squares, and T(k) a partition of k into cubes.

Links

Crossrefs

Cf. A000041, A001156, A003108, A369519, A369520, A369579.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/((1-x^k)*(1-x^(k^2))*(1-x^(k^3))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(2*n/3) + 3^(1/4) * zeta(3/2) * n^(1/4) / 2^(3/4) + 6^(1/6) * Gamma(4/3) * zeta(4/3) * n^(1/6) / Pi^(1/3) - 3*zeta(3/2)^2 / (32*Pi)) / (96 * Pi^(3/2) * n^(3/2)) * (1 - Gamma(1/3) * zeta(4/3) * zeta(3/2) / (12 * 6^(1/12) * Pi^(4/3) * n^(1/12))).

A385011 G.f.: 1/Product_{k>=1} (1 - x^(2*k^2)) * (1 - x^k).

Original entry on oeis.org

1, 1, 3, 4, 8, 11, 19, 26, 42, 57, 86, 116, 168, 224, 314, 415, 568, 743, 998, 1293, 1709, 2196, 2862, 3649, 4702, 5950, 7590, 9540, 12061, 15064, 18895, 23460, 29220, 36081, 44651, 54854, 67490, 82513, 100979, 122904, 149671, 181400, 219904, 265463, 320453, 385397
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 15 2025

Keywords

Comments

For n<=17, a(n-1) + a(n) = A369579(n).

Crossrefs

Cf. A000041, A001156, A369579, A385012.

Programs

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

Formula

a(n) ~ exp(Pi*sqrt(2*n/3) + 3^(1/4)*zeta(3/2)*n^(1/4)/2^(5/4) - 3*zeta(3/2)^2/(64*Pi)) / (2^(11/4) * 3^(3/4) * sqrt(Pi) * n^(5/4)).
Showing 1-5 of 5 results.

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