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 1-3 of 3 results.

A363068 Number of partitions p of n such that (1/5)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 59, 73, 94, 117, 148, 181, 228, 277, 344, 418, 514, 621, 762, 917, 1116, 1342, 1624, 1945, 2348, 2803, 3366, 4012, 4798, 5700, 6798, 8052, 9565, 11305, 13383, 15771, 18618, 21880, 25745, 30187, 35414, 41414, 48461, 56531, 65967
Offset: 0

Views

Author

Seiichi Manyama, May 16 2023

Keywords

Comments

In general, for m>=1, if g.f. = Sum_{k>=0} x^((m+1)*k) / Product_{j=1..m*k} (1 - x^j), then a(n) ~ Gamma(1/m) * Pi^(1/m) * exp(Pi*sqrt(2*n/3)) / (m^2 * 2^((4*m+1)/(2*m)) * 3^((m+1)/(2*m)) * n^(1 + 1/(2*m))). - Vaclav Kotesovec, Jun 19 2025

Examples

			a(8) = 2 counts these partitions:  521, 5111.

Links

Crossrefs

Cf. A002865, A238479, A363066, A363067.
Cf. A237827, A363047.
Cf. A035297, A035298.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^(6*k)/Product[1 - x^j, {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 18 2025 *)
nmax = 60; 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^(6*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)
  • PARI
    a(n) = sum(k=0, n\6, #partitions(n-6*k, 5*k));

Formula

G.f.: Sum_{k>=0} x^(6*k)/Product_{j=1..5*k} (1-x^j).
a(n) ~ Gamma(1/5) * Pi^(1/5) * exp(Pi*sqrt(2*n/3)) / (25 * 2^(21/10) * 3^(3/5) * n^(11/10)). - Vaclav Kotesovec, Jun 19 2025

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)).

A385092 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 15, 29, 53, 93, 155, 251, 397, 613, 929, 1385, 2033, 2945, 4219, 5979, 8393, 11683, 16133, 22119, 30125, 40773, 54867, 73435, 97785, 129583, 170941, 224519, 293673, 382615, 496609, 642231, 827667, 1063073, 1361029, 1737081, 2210381, 2804485, 3548303, 4477229
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

Comments

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

Links

Crossrefs

Cf. A035298, A385070.
Cf. A207641, A385088, A385089, A385090, A385091.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 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]]]; q = Expand[q*(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))]; 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(1/6) * exp(Pi*sqrt(n)) / (3 * 2^(10/3) * Pi^(5/6) * n^(7/12)).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.