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.

A327043 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k))).

Original entry on oeis.org

1, 1, 3, 5, 11, 16, 32, 47, 84, 124, 205, 298, 477, 681, 1044, 1484, 2211, 3097, 4516, 6261, 8948, 12295, 17273, 23511, 32597, 43975, 60187, 80601, 109114, 144999, 194423, 256584, 341008, 447178, 589558, 768398, 1005854, 1303450, 1694815, 2184666, 2823229
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 16 2019

Keywords

Comments

Differs from A006169.

Links

Crossrefs

Cf. A000041, A002513, A327042, A327044.
Cf. A006171.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 5^(5/2) * exp(5*Pi*sqrt(n/2)/3) / (288*2^(1/4)*n^(7/4)).

A327044 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(5*k))).

Original entry on oeis.org

1, 1, 3, 5, 11, 17, 33, 50, 89, 135, 223, 332, 530, 775, 1190, 1724, 2576, 3677, 5380, 7586, 10895, 15203, 21480, 29666, 41373, 56593, 77965, 105755, 144155, 193947, 261894, 349719, 468193, 620910, 824743, 1086661, 1433205, 1876865, 2459100, 3202155, 4170043
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 16 2019

Keywords

Comments

Differs from A006170.
In general, for fixed m>=1, if g.f. = Product_{k>=1} (Product_{j=1..m} 1/(1 - x^(j*k))), then a(n,m) ~ sqrt(Gamma(m+1)) * HarmonicNumber(m)^((m+1)/4) * exp(Pi*sqrt(2*HarmonicNumber(m)*n/3)) / (2^((3*m+5)/4) * 3^((m+1)/4) * n^((m+3)/4)).

Links

Crossrefs

Cf. A000041, A002513, A327042, A327043.
Cf. A006171.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 137^(3/2) * exp(sqrt(137*n/10)*Pi/3) / (2880*sqrt(6)*n^2).

A327045 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)).

Original entry on oeis.org

1, 1, 2, 4, 5, 8, 13, 17, 24, 36, 47, 64, 89, 115, 152, 204, 260, 336, 438, 552, 702, 896, 1117, 1400, 1758, 2171, 2688, 3332, 4079, 5000, 6131, 7446, 9048, 10992, 13255, 15984, 19264, 23081, 27644, 33084, 39408, 46912, 55797, 66107, 78264, 92572, 109140
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 16 2019

Keywords

Links

Crossrefs

Cf. A000009, A001935, A327046, A327047.
Cf. A107742, A327042.

Programs

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

Formula

a(n) ~ 11^(1/4) * exp(sqrt(11*n/2)*Pi/3) / (2^(13/4)*sqrt(3)*n^(3/4)).

A327048 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) / ((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k))).

Original entry on oeis.org

1, 2, 6, 14, 30, 60, 120, 220, 402, 710, 1224, 2064, 3438, 5596, 9012, 14304, 22422, 34740, 53330, 80960, 121908, 181976, 269484, 396072, 578232, 838258, 1207896, 1730058, 2463900, 3490020, 4918572, 6897012, 9626610, 13375776, 18504852, 25494456, 34985530
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 16 2019

Keywords

Comments

Convolution of A327045 and A327042.

Links

Crossrefs

Cf. A015128, A246584, A327049, A327050.
Cf. A301554.

Programs

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

Formula

a(n) ~ 11 * exp(sqrt(11*n/6)*Pi) / (2^(13/2)*sqrt(3)*n^(3/2)).
Showing 1-4 of 4 results.

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