A213424 Number of partitions of n in which all parts are >= 2 and the largest part occurs at least five times.

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 4, 6, 6, 8, 8, 12, 10, 15, 15, 19, 21, 26, 26, 34, 36, 45, 47, 59, 61, 76, 83, 97, 107, 128, 137, 165, 179, 210, 231, 271, 296, 345, 380, 438, 485, 561, 614, 708, 783, 893, 991, 1129, 1246, 1420, 1572, 1781

Offset: 10

Author

Mircea Merca, Jun 11 2012

Keywords

• nonn

Examples

			For n = 20 we have three partitions: {[4+4+4+4+4], [3+3+3+3+3+3+2], [2+2+2+2+2+2+2+2+2+2]}, so a(20) = 3.
		

Crossrefs

Cf. A000041, A186445, A213423.

Programs

Mathematica

nmax = 100; Drop[CoefficientList[Series[(1 - x)^2*(1 - x^2)*(1 - x^3)*(1 - x^4) / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x], 10] (* Vaclav Kotesovec, Jul 05 2025 *)

Formula

a(n) = A186445(n) - 2*A186445(n-1) + A186445(n-2).

G.f.: (1-x)*Product_{k>4} 1/(1-x^k).

a(n) ~ Pi^5 * exp(Pi*sqrt(2*n/3))/ (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jul 05 2025