A186445 The number of partitions of n in which the first part is at least four times larger than the second part.

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 20, 25, 31, 38, 47, 57, 70, 85, 103, 124, 150, 180, 216, 258, 308, 366, 436, 516, 611, 721, 850, 1000, 1176, 1378, 1614, 1886, 2203, 2567, 2990, 3474, 4034, 4677, 5417, 6264, 7239, 8351, 9628

Offset: 0

Author

Mircea Merca, Feb 21 2011

Keywords

• nonn

Examples

			a(8) = #{8, 7+1, 6+1+1, 5+1+1+1, 4+1+1+1+1} = 5.
a(10) = #{10, 9+1, 8+2, 8+1+1, 7+1+1+1, 6+1+1+1+1, 5+1+1+1+1+1, 4+1+1+1+1+1+1} = 8.
		

Crossrefs

Cf. A000041.

Partial sums of A185325.

Programs

Mathematica

Table[PartitionsP[n] - PartitionsP[n-2] - PartitionsP[n-3] - PartitionsP[n-4] + PartitionsP[n-5] + PartitionsP[n-6] + PartitionsP[n-7] - PartitionsP[n-9], {n,0,50}] (* Vaclav Kotesovec, Jul 05 2025 *)

Formula

a(n) = p(n)-p(n-2)-p(n-3)-p(n-4)+p(n-5)+p(n-6)+p(n-7)-p(n-9), where p(n) = A000041(n).

a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3*sqrt(2)*n^(5/2)) * (1 - (5*sqrt(6)/Pi + 109*Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jul 05 2025