A035296 -id:A035296 - cp's OEIS frontend

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

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

Vaclav Kotesovec, Jun 16 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

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

Cf. A035296.

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]

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

A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 39, 51, 64, 81, 102, 128, 159, 198, 245, 304, 374, 460, 563, 689, 841, 1023, 1242, 1505, 1819, 2195, 2642, 3173, 3804, 4551, 5435, 6477, 7707, 9151, 10850, 12843, 15175, 17902, 21089, 24802, 29132, 34164, 40012, 46796, 54663, 63766

Offset: 0

Seiichi Manyama, May 16 2023

nonn

			a(8) = 3 counts these partitions:  431, 4211, 41111.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Cf. A002865, A238479, A363066, A363068.
Cf. A237826, A363046.
Cf. A035296.

nmax = 60; CoefficientList[Series[Sum[x^(5*k)/Product[1 - x^j, {j, 1, 4*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^(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+=x^(5*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)

a(n) = sum(k=0, n\5, #partitions(n-5*k, 4*k));

G.f.: Sum_{k>=0} x^(5*k)/Product_{j=1..4*k} (1-x^j).

a(n) ~ Gamma(1/4) * Pi^(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(49/8) * 3^(5/8) * n^(9/8)). - Vaclav Kotesovec, Jun 19 2025

A385090 G.f.: Sum_{k>=0} x^k * Product_{j=1..4*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 15, 29, 51, 87, 143, 227, 353, 537, 803, 1185, 1727, 2489, 3551, 5021, 7039, 9791, 13521, 18541, 25261, 34207, 46051, 61655, 82113, 108815, 143517, 188433, 246343, 320725, 415931, 537377, 691791, 887517, 1134863, 1446549, 1838235, 2329147, 2942849, 3708165

Offset: 0

Vaclav Kotesovec, Jun 17 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A035296, A385068.

Cf. A207641, A385088, A385089, A385091, A385092.

nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 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]]]; q = Expand[q*(1 + x^(4*k))*(1 + x^(4*k - 1))*(1 + x^(4*k - 2))*(1 + x^(4*k - 3))]; 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]

a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n)) / (2^(9/2) * Pi^(3/4) * n^(5/8)).

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

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A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.

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A385090 G.f.: Sum_{k>=0} x^k * Product_{j=1..4*k} (1 + x^j)/(1 - x^j).

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