A035295 -id:A035295 - cp's OEIS frontend

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

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

Vaclav Kotesovec, Jun 16 2025

nonn

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

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

Cf. A035295.

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]

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

A363066 Number of partitions p of n such that (1/3)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 16, 20, 27, 33, 45, 55, 72, 89, 116, 142, 181, 222, 281, 343, 429, 522, 649, 786, 967, 1168, 1429, 1719, 2088, 2504, 3026, 3615, 4345, 5174, 6192, 7349, 8755, 10360, 12297, 14507, 17154, 20182, 23788, 27910, 32790, 38374, 44955, 52480, 61307, 71402

Offset: 0

Seiichi Manyama, May 16 2023

nonn

			a(7) = 3 counts these partitions:  331, 3211, 31111.

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

Cf. A002865, A238479, A363067, A363068.
Cf. A008484, A237825, A363045.
Cf. A035295.

nmax = 60; CoefficientList[Series[Sum[x^(4*k)/Product[1 - x^j, {j, 1, 3*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^(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+=x^(4*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)
Join[{1},Table[Count[IntegerPartitions[n],?(MemberQ[#,#[[1]]/3]&)],{n,60}]] (* _Harvey P. Dale, Jun 29 2025 *)

a(n) = sum(k=0, n\4, #partitions(n-4*k, 3*k));

G.f.: Sum_{k>=0} x^(4*k)/Product_{j=1..3*k} (1-x^j).

a(n) ~ Gamma(1/3) * Pi^(1/3) * exp(Pi*sqrt(2*n/3)) / (2^(13/6) * 3^(8/3) * n^(7/6)). - Vaclav Kotesovec, Jun 19 2025

A385089 G.f.: Sum_{k>=0} x^k * Product_{j=1..3*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 15, 27, 47, 79, 127, 199, 307, 465, 695, 1025, 1493, 2151, 3069, 4337, 6075, 8441, 11639, 15933, 21667, 29281, 39337, 52555, 69849, 92375, 121595, 159347, 207939, 270259, 349911, 451377, 580223, 743341, 949241, 1208415, 1533763, 1941111, 2449841, 3083637

Offset: 0

Vaclav Kotesovec, Jun 17 2025

nonn

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

Cf. A035295, A385067.

Cf. A207641, A385088, A385090, A385091, A385092.

nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 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]]]; q = Expand[q*(1 + x^(3*k))*(1 + x^(3*k - 1))*(1 + x^(3*k - 2))]; 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(1/3) * exp(Pi*sqrt(n)) / (3 * 2^(8/3) * Pi^(2/3) * n^(2/3)).

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

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A363066 Number of partitions p of n such that (1/3)*max(p) is a part of p.

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A385089 G.f.: Sum_{k>=0} x^k * Product_{j=1..3*k} (1 + x^j)/(1 - x^j).

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