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 1-4 of 4 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).

A280276 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^2)).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 17, 24, 33, 46, 62, 82, 108, 141, 182, 233, 297, 375, 472, 590, 733, 907, 1117, 1369, 1671, 2034, 2465, 2978, 3586, 4304, 5152, 6149, 7319, 8689, 10293, 12162, 14340, 16871, 19806, 23207, 27139, 31678, 36909, 42932, 49851, 57794, 66897
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 30 2016

Keywords

Comments

Convolution of A000009 and A001156.

Links

Crossrefs

Cf. A000009, A001156, A087154, A103265, A280277, A369570, A369574.

Programs

  • Mathematica
    nmax=80; CoefficientList[Series[Product[(1+x^k)/(1-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(n/3) + 3^(1/4) * Zeta(3/2) * n^(1/4) / sqrt(2) - 3*Zeta(3/2)^2 / (16*Pi)) / (8*sqrt(6*Pi)*n).

A369578 a(n) = [x^n] Product_{j=1..n, k=1..n} (1 + x^(k^j)).

Original entry on oeis.org

1, 1, 2, 5, 13, 33, 81, 194, 458, 1074, 2513, 5876, 13722, 31961, 74168, 171395, 394450, 904393, 2066770, 4709538, 10704306, 24273709, 54926658, 124036675, 279559571, 628906790, 1412254773, 3165780760, 7084607367, 15828666526, 35309625162, 78648201835, 174927430681
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

Links

Crossrefs

Cf. A000009, A033461, A279329, A369575, A369570.

Programs

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

A369575 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(k^2)) * (1 + x^(k^3)).

Original entry on oeis.org

1, 3, 4, 5, 8, 12, 16, 21, 28, 38, 51, 65, 82, 105, 133, 166, 206, 254, 312, 382, 464, 561, 677, 813, 972, 1160, 1380, 1636, 1935, 2281, 2682, 3148, 3683, 4297, 5008, 5826, 6761, 7832, 9055, 10451, 12045, 13855, 15909, 18246, 20895, 23891, 27282, 31110, 35427
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

Comments

Convolution of A000009 and A033461 and A279329.
Convolution of A369570 and A279329.
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 into distinct parts, S(k) a partition of k into distinct squares, and T(k) a partition of k into distinct cubes.

Links

Crossrefs

Cf. A000009, A033461, A279329, A369570, A369571, A369572.

Programs

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

Formula

a(n) ~ exp(Pi*sqrt(n/3) + (2^(1/3) - 1) * Gamma(1/3) * zeta(4/3) * n^(1/6) / (3^(5/6) * Pi^(1/3)) + 3^(1/4)*(sqrt(2) - 1) * zeta(3/2) * n^(1/4)/2 + 3*(2*sqrt(2) - 3) * zeta(3/2)^2 / (32*Pi)) / (8*3^(1/4)*n^(3/4)).
Showing 1-4 of 4 results.

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

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