A035297 -id:A035297 - cp's OEIS frontend

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

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

Seiichi Manyama, May 16 2023

nonn

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

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Seiichi Manyama)

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

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 *)

a(n) = sum(k=0, n\6, #partitions(n-6*k, 5*k));

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

A385069 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 36, 45, 56, 69, 86, 106, 130, 159, 192, 232, 279, 333, 397, 472, 557, 657, 773, 905, 1059, 1236, 1437, 1669, 1935, 2236, 2582, 2976, 3422, 3930, 4507, 5157, 5894, 6728, 7664, 8721, 9911, 11246, 12746, 14430, 16312, 18421, 20780

Offset: 0

Vaclav Kotesovec, Jun 16 2025

nonn

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

Cf. A000009, A192433, A385067, A385068, A385070.

Cf. A035297.

nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]
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 += p*x^k;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

a(n) ~ Gamma(1/5) * 3^(3/20) * exp(Pi*sqrt(n/3)) / (5 * 2^(6/5) * Pi^(4/5) * n^(7/20)).

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

Vaclav Kotesovec, Jun 17 2025

nonn

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

Cf. A035297, A385069.

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

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]

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

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

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A385069 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j).

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

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