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.

A385067 G.f.: Sum_{k>=0} x^k * Product_{j=1..3*k} (1 + x^j).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 11, 14, 18, 23, 30, 38, 47, 58, 71, 87, 106, 128, 154, 185, 221, 263, 313, 370, 437, 514, 603, 705, 822, 958, 1112, 1289, 1491, 1721, 1982, 2279, 2617, 2999, 3432, 3921, 4473, 5095, 5795, 6583, 7468, 8461, 9574, 10820, 12214, 13772, 15512, 17453
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 16 2025

Keywords

Links

Crossrefs

Cf. A000009, A192433, A385068, A385069, A385070.
Cf. A035295.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Expand[p*(1 + x^(3*k))*(1 + x^(3*k - 1))*(1 + x^(3*k - 2))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p*x^k;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/3) * exp(Pi*sqrt(n/3)) / (2^(4/3) * 3^(11/12) * Pi^(2/3) * n^(5/12)).

A385068 G.f.: Sum_{k>=0} x^k * Product_{j=1..4*k} (1 + x^j).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 26, 33, 41, 52, 65, 81, 99, 121, 147, 177, 214, 255, 304, 362, 429, 507, 596, 700, 820, 959, 1119, 1301, 1510, 1750, 2023, 2335, 2688, 3089, 3546, 4062, 4647, 5306, 6050, 6889, 7833, 8895, 10085, 11422, 12921, 14599, 16477, 18573, 20914
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 16 2025

Keywords

Links

Crossrefs

Cf. A000009, A192433, A385067, A385069, A385070.
Cf. A035296.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Expand[p*(1 + x^(4*k))*(1 + x^(4*k - 1))*(1 + x^(4*k - 2))*(1 + x^(4*k - 3))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p*x^k;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/4) * 3^(1/8) * exp(Pi*sqrt(n/3)) / (2^(13/4) * Pi^(3/4) * n^(3/8)).

A385070 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 18, 23, 30, 38, 48, 60, 74, 91, 112, 137, 166, 202, 244, 294, 352, 420, 500, 592, 700, 824, 968, 1133, 1323, 1541, 1791, 2077, 2403, 2776, 3198, 3679, 4226, 4845, 5546, 6340, 7236, 8246, 9385, 10667, 12108, 13728, 15545, 17581, 19860, 22409
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 16 2025

Keywords

Comments

In general, for m>=1, if g.f. = Sum_{k>=0} x^k * Product_{j=1..m*k} (1 + x^j), then a(n) ~ Gamma(1/m) * 3^((m-2)/(4*m)) * exp(Pi*sqrt(n/3)) / (m * 2^(1 + 1/m) * Pi^(1 - 1/m) * n^((m+2)/(4*m))). - Vaclav Kotesovec, Jun 17 2025

Links

Crossrefs

Cf. A000009, A192433, A385067, A385068, A385069.
Cf. A035298.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 60; p = 1; s = 1; Do[p = Expand[p*(1 + x^(6*k))*(1 + x^(6*k - 1))*(1 + x^(6*k - 2))*(1 + x^(6*k - 3))*(1 + x^(6*k - 4))*(1 + x^(6*k - 5))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p*x^k;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/6) * exp(Pi*sqrt(n/3)) / (2^(13/6) * 3^(5/6) * Pi^(5/6) * n^(1/3)).

A385091 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 15, 29, 53, 91, 151, 243, 381, 585, 881, 1305, 1907, 2753, 3931, 5559, 7793, 10835, 14955, 20501, 27921, 37801, 50889, 68139, 90777, 120353, 158827, 208683, 273037, 355791, 461839, 597273, 769661, 988411, 1265149, 1614215, 2053297, 2604113, 3293281, 4153407
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

Links

Crossrefs

Cf. A035297, A385069.
Cf. A207641, A385088, A385089, A385090, A385092.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(5*k))*(1 - x^(5*k - 1))*(1 - x^(5*k - 2))*(1 - x^(5*k - 3))*(1 - x^(5*k - 4))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(5*k))*(1 + x^(5*k - 1))*(1 + x^(5*k - 2))*(1 + x^(5*k - 3))*(1 + x^(5*k - 4))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/5) * exp(Pi*sqrt(n)) / (5 * 2^(12/5) * Pi^(4/5) * n^(3/5)).

A035297 Expansion of sum ( q^n / product( 1-q^k, k=1..5*n), n=0..inf ).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 19, 29, 43, 63, 90, 127, 176, 241, 327, 439, 585, 773, 1015, 1322, 1714, 2208, 2831, 3610, 4585, 5794, 7297, 9149, 11433, 14233, 17665, 21846, 26943, 33123, 40614, 49656, 60565, 73671, 89414, 108254, 130785, 157649, 189654, 227671, 272802, 326236, 389446
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A363068, A385069.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 16 2025 *)
nmax = 50; p=1; s=1; Do[p=Expand[p*(1-x^(5*k))*(1-x^(5*k-1))*(1-x^(5*k-2))*(1-x^(5*k-3))*(1-x^(5*k-4))];p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]];s+=x^k/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2025 *)

Formula

a(n) ~ Gamma(1/5) * exp(Pi*sqrt(2*n/3)) / (5 * 2^(8/5) * 3^(1/10) * Pi^(4/5) * n^(3/5)). - Vaclav Kotesovec, Jun 17 2025

Extensions

More terms from Vaclav Kotesovec, Jun 16 2025
Showing 1-5 of 5 results.

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