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

Author

Mircea Merca, Jun 11 2012

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A000041.

Programs

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)

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