A213423 Number of partitions of n in which all parts are >= 2 and the largest part occurs at least four times.

1, 0, 1, 0, 2, 0, 2, 1, 3, 1, 4, 2, 6, 3, 7, 5, 11, 7, 13, 11, 19, 15, 25, 21, 34, 30, 44, 42, 60, 56, 78, 78, 105, 103, 137, 139, 181, 186, 234, 246, 309, 323, 399, 425, 519, 554, 670, 721, 864, 934, 1108, 1206, 1425, 1548, 1816, 1989, 2318, 2539, 2945, 3235, 3738, 4111, 4726

Offset

8,5

Links

Table of n, a(n) for n=8..70.

Formula

a(n) = p(n)-2*p(n-1)+p(n-3)+p(n-4)-2*p(n-6)+p(n-7), where p(n) = A000041(n).

G.f.: (1-x)*Product_{k>3} 1/(1-x^k).

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (24*sqrt(3)*n^3). - Vaclav Kotesovec, Jun 02 2018

G.f.: Sum_{n >= 1} q^(4*n+4)/Product_{k = 1..n} 1- q^(k+1). - Peter Bala, Dec 01 2024

Example

For n = 16 we have three partitions: {[4+4+4+4], [3+3+3+3+2+2], [2+2+2+2+2+2+2+2]}, so a(16) = 3.

Maple

seq(combinat:-numbpart(n)-2*combinat:-numbpart(n-1)+combinat:-numbpart(n-3)+combinat:-numbpart(n-4)-2*combinat:-numbpart(n-6)+combinat:-numbpart(n-7), n=8..70)

Crossrefs

Cf. A000041.

Sequence in context: A029221 A304034 A029183 * A339374 A265753 A138110

Adjacent sequences: A213420 A213421 A213422 * A213424 A213425 A213426

Keyword

nonn

Author

Mircea Merca, Jun 11 2012

Status

approved