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.

A280204 G.f.: Product_{k>=1} (1+x^(k^2)) / (1-x^k).

Original entry on oeis.org

1, 2, 3, 5, 9, 14, 21, 31, 45, 65, 92, 127, 175, 239, 322, 430, 572, 753, 985, 1281, 1657, 2131, 2727, 3471, 4401, 5558, 6988, 8751, 10924, 13588, 16846, 20819, 25653, 31518, 38621, 47195, 57530, 69958, 84869, 102723, 124070, 149532, 179852, 215894, 258668
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 28 2016

Keywords

Comments

Convolution of A033461 and A000041.

Links

Crossrefs

Cf. A000041, A033461, A052335, A078434, A087153, A117144, A264393, A280224, A280225, A369520, A369570.

Programs

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

Formula

a(n) ~ exp(Pi*sqrt(2*n/3) + 2^(-5/4)*3^(1/4)*(sqrt(2)-1)*Zeta(3/2)*n^(1/4) - 3*(sqrt(2)-1)^2*Zeta(3/2)^2/(64*Pi)) / (2^(5/2)*sqrt(3)*n).

A369579 Expansion of Product_{k>=1} 1 / ((1 - x^k) * (1 - x^(k^3))).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 68, 99, 143, 202, 284, 392, 538, 729, 983, 1311, 1740, 2289, 2998, 3898, 5046, 6492, 8321, 10607, 13472, 17032, 21460, 26927, 33682, 41975, 52160, 64600, 79790, 98255, 120690, 147836, 180662, 220217, 267841, 324999, 393539, 475496, 573403
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

Comments

Convolution of A000041 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 and P(n-k) is a partition of n-k into cubes.

Links

Crossrefs

Cf. A000041, A001156, A003108, A280278, A369519, A369520.

Programs

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

Formula

a(n) ~ exp(Pi*sqrt(2*n/3) + 6^(1/6) * Gamma(4/3) * zeta(4/3) * n^(1/6) / Pi^(1/3)) / (2^(15/4) * 3^(3/4) * Pi * n^(5/4)) * (1 - Gamma(1/3)^2 * zeta(4/3)^2 / (54 * 6^(1/6) * Pi^(5/3) * n^(1/6))).

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))).

A369577 a(n) = [x^n] Product_{j=1..n, k=1..n} 1/(1 - x^(k^j)).

Original entry on oeis.org

1, 1, 4, 14, 52, 193, 724, 2736, 10404, 39759, 152555, 587323, 2267578, 8776197, 34038411, 132262696, 514774705, 2006461961, 7830924282, 30599035846, 119692591204, 468651774760, 1836626054421, 7203559635483, 28274941506056, 111060542576799, 436515284729667
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

Links

Crossrefs

Cf. A000041, A001156, A003108, A369520, A369576.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[Product[1/(1 - x^(k^j)), {k, 1, n^(1/j)}], {j, 1, n}], {x, 0, n}], {n, 0, 40}]

Formula

a(n) ~ c * 4^n / sqrt(n), where c = 0.52405470637768487694539405770364130415279761385131429278498764796443...
Showing 1-5 of 5 results.

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