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

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

0,7

Links

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

Formula

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

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A002865, A238479, A363067, A363068.

Cf. A008484, A237825, A363045.

Cf. A035295.

Sequence in context: A027588 A224956 A131995 * A060714 A241819 A227070

Adjacent sequences: A363063 A363064 A363065 * A363067 A363068 A363069

Keyword

nonn

Author

Seiichi Manyama, May 16 2023

Status

approved